Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of 1/5x
Integral of (sec(x))^3
Integral of lnx/sqrtx
Integral of x^2*e^(x^3)
Identical expressions
cos^27x
co sinus of e of squared 7x
cos27x
cos²7x
cos to the power of 27x
cos^27xdx

Integral of cos^27x dx

The solution

You have entered [src]

 oo            
  /            
 |             
 |     27      
 |  cos  (x) dx
 |             
/              
2

$$\int\limits_{2}^{\infty} \cos^{27}{\left(x \right)}\, dx$$

Integral(cos(x)^27, (x, 2, oo))

Detail solution

Rewrite the integrand:

$\cos^{27}{\left(x \right)} = \left(1 - \sin^{2}{\left(x \right)}\right)^{13} \cos{\left(x \right)}$
There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\left(1 - \sin^{2}{\left(x \right)}\right)^{13} \cos{\left(x \right)} = - \sin^{26}{\left(x \right)} \cos{\left(x \right)} + 13 \sin^{24}{\left(x \right)} \cos{\left(x \right)} - 78 \sin^{22}{\left(x \right)} \cos{\left(x \right)} + 286 \sin^{20}{\left(x \right)} \cos{\left(x \right)} - 715 \sin^{18}{\left(x \right)} \cos{\left(x \right)} + 1287 \sin^{16}{\left(x \right)} \cos{\left(x \right)} - 1716 \sin^{14}{\left(x \right)} \cos{\left(x \right)} + 1716 \sin^{12}{\left(x \right)} \cos{\left(x \right)} - 1287 \sin^{10}{\left(x \right)} \cos{\left(x \right)} + 715 \sin^{8}{\left(x \right)} \cos{\left(x \right)} - 286 \sin^{6}{\left(x \right)} \cos{\left(x \right)} + 78 \sin^{4}{\left(x \right)} \cos{\left(x \right)} - 13 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \sin^{26}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin^{26}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{26}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{26}\, du = \frac{u^{27}}{27}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{27}{\left(x \right)}}{27}$
    So, the result is: $- \frac{\sin^{27}{\left(x \right)}}{27}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 13 \sin^{24}{\left(x \right)} \cos{\left(x \right)}\, dx = 13 \int \sin^{24}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{24}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{24}\, du = \frac{u^{25}}{25}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{25}{\left(x \right)}}{25}$
    So, the result is: $\frac{13 \sin^{25}{\left(x \right)}}{25}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 78 \sin^{22}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 78 \int \sin^{22}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{22}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{22}\, du = \frac{u^{23}}{23}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{23}{\left(x \right)}}{23}$
    So, the result is: $- \frac{78 \sin^{23}{\left(x \right)}}{23}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 286 \sin^{20}{\left(x \right)} \cos{\left(x \right)}\, dx = 286 \int \sin^{20}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{20}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{20}\, du = \frac{u^{21}}{21}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{21}{\left(x \right)}}{21}$
    So, the result is: $\frac{286 \sin^{21}{\left(x \right)}}{21}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 715 \sin^{18}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 715 \int \sin^{18}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{18}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{18}\, du = \frac{u^{19}}{19}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{19}{\left(x \right)}}{19}$
    So, the result is: $- \frac{715 \sin^{19}{\left(x \right)}}{19}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 1287 \sin^{16}{\left(x \right)} \cos{\left(x \right)}\, dx = 1287 \int \sin^{16}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{16}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{16}\, du = \frac{u^{17}}{17}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{17}{\left(x \right)}}{17}$
    So, the result is: $\frac{1287 \sin^{17}{\left(x \right)}}{17}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 1716 \sin^{14}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 1716 \int \sin^{14}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{14}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{14}\, du = \frac{u^{15}}{15}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{15}{\left(x \right)}}{15}$
    So, the result is: $- \frac{572 \sin^{15}{\left(x \right)}}{5}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 1716 \sin^{12}{\left(x \right)} \cos{\left(x \right)}\, dx = 1716 \int \sin^{12}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{12}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{12}\, du = \frac{u^{13}}{13}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{13}{\left(x \right)}}{13}$
    So, the result is: $132 \sin^{13}{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 1287 \sin^{10}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 1287 \int \sin^{10}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{10}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{10}\, du = \frac{u^{11}}{11}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{11}{\left(x \right)}}{11}$
    So, the result is: $- 117 \sin^{11}{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 715 \sin^{8}{\left(x \right)} \cos{\left(x \right)}\, dx = 715 \int \sin^{8}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{8}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{8}\, du = \frac{u^{9}}{9}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{9}{\left(x \right)}}{9}$
    So, the result is: $\frac{715 \sin^{9}{\left(x \right)}}{9}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 286 \sin^{6}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 286 \int \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{6}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{6}\, du = \frac{u^{7}}{7}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{7}{\left(x \right)}}{7}$
    So, the result is: $- \frac{286 \sin^{7}{\left(x \right)}}{7}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 78 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx = 78 \int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{4}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{4}\, du = \frac{u^{5}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{5}{\left(x \right)}}{5}$
    So, the result is: $\frac{78 \sin^{5}{\left(x \right)}}{5}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 13 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 13 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{3}{\left(x \right)}}{3}$
    So, the result is: $- \frac{13 \sin^{3}{\left(x \right)}}{3}$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  The result is: $- \frac{\sin^{27}{\left(x \right)}}{27} + \frac{13 \sin^{25}{\left(x \right)}}{25} - \frac{78 \sin^{23}{\left(x \right)}}{23} + \frac{286 \sin^{21}{\left(x \right)}}{21} - \frac{715 \sin^{19}{\left(x \right)}}{19} + \frac{1287 \sin^{17}{\left(x \right)}}{17} - \frac{572 \sin^{15}{\left(x \right)}}{5} + 132 \sin^{13}{\left(x \right)} - 117 \sin^{11}{\left(x \right)} + \frac{715 \sin^{9}{\left(x \right)}}{9} - \frac{286 \sin^{7}{\left(x \right)}}{7} + \frac{78 \sin^{5}{\left(x \right)}}{5} - \frac{13 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$
Method #2
1. Rewrite the integrand:
  
  $\left(1 - \sin^{2}{\left(x \right)}\right)^{13} \cos{\left(x \right)} = - \sin^{26}{\left(x \right)} \cos{\left(x \right)} + 13 \sin^{24}{\left(x \right)} \cos{\left(x \right)} - 78 \sin^{22}{\left(x \right)} \cos{\left(x \right)} + 286 \sin^{20}{\left(x \right)} \cos{\left(x \right)} - 715 \sin^{18}{\left(x \right)} \cos{\left(x \right)} + 1287 \sin^{16}{\left(x \right)} \cos{\left(x \right)} - 1716 \sin^{14}{\left(x \right)} \cos{\left(x \right)} + 1716 \sin^{12}{\left(x \right)} \cos{\left(x \right)} - 1287 \sin^{10}{\left(x \right)} \cos{\left(x \right)} + 715 \sin^{8}{\left(x \right)} \cos{\left(x \right)} - 286 \sin^{6}{\left(x \right)} \cos{\left(x \right)} + 78 \sin^{4}{\left(x \right)} \cos{\left(x \right)} - 13 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \sin^{26}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin^{26}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{26}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{26}\, du = \frac{u^{27}}{27}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{27}{\left(x \right)}}{27}$
    So, the result is: $- \frac{\sin^{27}{\left(x \right)}}{27}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 13 \sin^{24}{\left(x \right)} \cos{\left(x \right)}\, dx = 13 \int \sin^{24}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{24}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{24}\, du = \frac{u^{25}}{25}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{25}{\left(x \right)}}{25}$
    So, the result is: $\frac{13 \sin^{25}{\left(x \right)}}{25}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 78 \sin^{22}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 78 \int \sin^{22}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{22}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{22}\, du = \frac{u^{23}}{23}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{23}{\left(x \right)}}{23}$
    So, the result is: $- \frac{78 \sin^{23}{\left(x \right)}}{23}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 286 \sin^{20}{\left(x \right)} \cos{\left(x \right)}\, dx = 286 \int \sin^{20}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{20}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{20}\, du = \frac{u^{21}}{21}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{21}{\left(x \right)}}{21}$
    So, the result is: $\frac{286 \sin^{21}{\left(x \right)}}{21}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 715 \sin^{18}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 715 \int \sin^{18}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{18}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{18}\, du = \frac{u^{19}}{19}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{19}{\left(x \right)}}{19}$
    So, the result is: $- \frac{715 \sin^{19}{\left(x \right)}}{19}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 1287 \sin^{16}{\left(x \right)} \cos{\left(x \right)}\, dx = 1287 \int \sin^{16}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{16}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{16}\, du = \frac{u^{17}}{17}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{17}{\left(x \right)}}{17}$
    So, the result is: $\frac{1287 \sin^{17}{\left(x \right)}}{17}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 1716 \sin^{14}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 1716 \int \sin^{14}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{14}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{14}\, du = \frac{u^{15}}{15}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{15}{\left(x \right)}}{15}$
    So, the result is: $- \frac{572 \sin^{15}{\left(x \right)}}{5}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 1716 \sin^{12}{\left(x \right)} \cos{\left(x \right)}\, dx = 1716 \int \sin^{12}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{12}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{12}\, du = \frac{u^{13}}{13}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{13}{\left(x \right)}}{13}$
    So, the result is: $132 \sin^{13}{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 1287 \sin^{10}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 1287 \int \sin^{10}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{10}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{10}\, du = \frac{u^{11}}{11}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{11}{\left(x \right)}}{11}$
    So, the result is: $- 117 \sin^{11}{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 715 \sin^{8}{\left(x \right)} \cos{\left(x \right)}\, dx = 715 \int \sin^{8}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{8}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{8}\, du = \frac{u^{9}}{9}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{9}{\left(x \right)}}{9}$
    So, the result is: $\frac{715 \sin^{9}{\left(x \right)}}{9}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 286 \sin^{6}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 286 \int \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{6}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{6}\, du = \frac{u^{7}}{7}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{7}{\left(x \right)}}{7}$
    So, the result is: $- \frac{286 \sin^{7}{\left(x \right)}}{7}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 78 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx = 78 \int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{4}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{4}\, du = \frac{u^{5}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{5}{\left(x \right)}}{5}$
    So, the result is: $\frac{78 \sin^{5}{\left(x \right)}}{5}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 13 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 13 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{3}{\left(x \right)}}{3}$
    So, the result is: $- \frac{13 \sin^{3}{\left(x \right)}}{3}$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  The result is: $- \frac{\sin^{27}{\left(x \right)}}{27} + \frac{13 \sin^{25}{\left(x \right)}}{25} - \frac{78 \sin^{23}{\left(x \right)}}{23} + \frac{286 \sin^{21}{\left(x \right)}}{21} - \frac{715 \sin^{19}{\left(x \right)}}{19} + \frac{1287 \sin^{17}{\left(x \right)}}{17} - \frac{572 \sin^{15}{\left(x \right)}}{5} + 132 \sin^{13}{\left(x \right)} - 117 \sin^{11}{\left(x \right)} + \frac{715 \sin^{9}{\left(x \right)}}{9} - \frac{286 \sin^{7}{\left(x \right)}}{7} + \frac{78 \sin^{5}{\left(x \right)}}{5} - \frac{13 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$
Now simplify:

$\frac{\left(- 1300075 \sin^{26}{\left(x \right)} + 18253053 \sin^{24}{\left(x \right)} - 119041650 \sin^{22}{\left(x \right)} + 478056150 \sin^{20}{\left(x \right)} - 1320944625 \sin^{18}{\left(x \right)} + 2657429775 \sin^{16}{\left(x \right)} - 4015671660 \sin^{14}{\left(x \right)} + 4633467300 \sin^{12}{\left(x \right)} - 4106936925 \sin^{10}{\left(x \right)} + 2788660875 \sin^{8}{\left(x \right)} - 1434168450 \sin^{6}{\left(x \right)} + 547591590 \sin^{4}{\left(x \right)} - 152108775 \sin^{2}{\left(x \right)} + 35102025\right) \sin{\left(x \right)}}{35102025}$
Add the constant of integration:

$\frac{\left(- 1300075 \sin^{26}{\left(x \right)} + 18253053 \sin^{24}{\left(x \right)} - 119041650 \sin^{22}{\left(x \right)} + 478056150 \sin^{20}{\left(x \right)} - 1320944625 \sin^{18}{\left(x \right)} + 2657429775 \sin^{16}{\left(x \right)} - 4015671660 \sin^{14}{\left(x \right)} + 4633467300 \sin^{12}{\left(x \right)} - 4106936925 \sin^{10}{\left(x \right)} + 2788660875 \sin^{8}{\left(x \right)} - 1434168450 \sin^{6}{\left(x \right)} + 547591590 \sin^{4}{\left(x \right)} - 152108775 \sin^{2}{\left(x \right)} + 35102025\right) \sin{\left(x \right)}}{35102025}+ \mathrm{constant}$

The answer is:

$\frac{\left(- 1300075 \sin^{26}{\left(x \right)} + 18253053 \sin^{24}{\left(x \right)} - 119041650 \sin^{22}{\left(x \right)} + 478056150 \sin^{20}{\left(x \right)} - 1320944625 \sin^{18}{\left(x \right)} + 2657429775 \sin^{16}{\left(x \right)} - 4015671660 \sin^{14}{\left(x \right)} + 4633467300 \sin^{12}{\left(x \right)} - 4106936925 \sin^{10}{\left(x \right)} + 2788660875 \sin^{8}{\left(x \right)} - 1434168450 \sin^{6}{\left(x \right)} + 547591590 \sin^{4}{\left(x \right)} - 152108775 \sin^{2}{\left(x \right)} + 35102025\right) \sin{\left(x \right)}}{35102025}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                                                                                                                                
 |                                                        19             15             7            23            3         27            25            5             21             9              17            
 |    27                    11             13      715*sin  (x)   572*sin  (x)   286*sin (x)   78*sin  (x)   13*sin (x)   sin  (x)   13*sin  (x)   78*sin (x)   286*sin  (x)   715*sin (x)   1287*sin  (x)         
 | cos  (x) dx = C - 117*sin  (x) + 132*sin  (x) - ------------ - ------------ - ----------- - ----------- - ---------- - -------- + ----------- + ---------- + ------------ + ----------- + ------------- + sin(x)
 |                                                      19             5              7             23           3           27           25           5             21             9              17              
/

$$\int \cos^{27}{\left(x \right)}\, dx = C - \frac{\sin^{27}{\left(x \right)}}{27} + \frac{13 \sin^{25}{\left(x \right)}}{25} - \frac{78 \sin^{23}{\left(x \right)}}{23} + \frac{286 \sin^{21}{\left(x \right)}}{21} - \frac{715 \sin^{19}{\left(x \right)}}{19} + \frac{1287 \sin^{17}{\left(x \right)}}{17} - \frac{572 \sin^{15}{\left(x \right)}}{5} + 132 \sin^{13}{\left(x \right)} - 117 \sin^{11}{\left(x \right)} + \frac{715 \sin^{9}{\left(x \right)}}{9} - \frac{286 \sin^{7}{\left(x \right)}}{7} + \frac{78 \sin^{5}{\left(x \right)}}{5} - \frac{13 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$$

Simplify

The answer [src]

                                                                17             9             21            5            25         27            3            23             7             15             19                                                                  17             9             21            5            25         27            3            23             7             15             19    
   22308732928                   13             11      1287*sin  (2)   715*sin (2)   286*sin  (2)   78*sin (2)   13*sin  (2)   sin  (2)   13*sin (2)   78*sin  (2)   286*sin (2)   572*sin  (2)   715*sin  (2)  22308732928                   13             11      1287*sin  (2)   715*sin (2)   286*sin  (2)   78*sin (2)   13*sin  (2)   sin  (2)   13*sin (2)   78*sin  (2)   286*sin (2)   572*sin  (2)   715*sin  (2) 
<- ----------- - sin(2) - 132*sin  (2) + 117*sin  (2) - ------------- - ----------- - ------------ - ---------- - ----------- + -------- + ---------- + ----------- + ----------- + ------------ + ------------, ----------- - sin(2) - 132*sin  (2) + 117*sin  (2) - ------------- - ----------- - ------------ - ---------- - ----------- + -------- + ---------- + ----------- + ----------- + ------------ + ------------>
     35102025                                                 17             9             21            5             25          27          3             23            7             5              19         35102025                                                 17             9             21            5             25          27          3             23            7             5              19

$$\left\langle - \frac{22308732928}{35102025} - 132 \sin^{13}{\left(2 \right)} - \frac{715 \sin^{9}{\left(2 \right)}}{9} - \frac{1287 \sin^{17}{\left(2 \right)}}{17} - \frac{78 \sin^{5}{\left(2 \right)}}{5} - \frac{286 \sin^{21}{\left(2 \right)}}{21} - \sin{\left(2 \right)} - \frac{13 \sin^{25}{\left(2 \right)}}{25} + \frac{\sin^{27}{\left(2 \right)}}{27} + \frac{78 \sin^{23}{\left(2 \right)}}{23} + \frac{13 \sin^{3}{\left(2 \right)}}{3} + \frac{715 \sin^{19}{\left(2 \right)}}{19} + \frac{286 \sin^{7}{\left(2 \right)}}{7} + \frac{572 \sin^{15}{\left(2 \right)}}{5} + 117 \sin^{11}{\left(2 \right)}, - 132 \sin^{13}{\left(2 \right)} - \frac{715 \sin^{9}{\left(2 \right)}}{9} - \frac{1287 \sin^{17}{\left(2 \right)}}{17} - \frac{78 \sin^{5}{\left(2 \right)}}{5} - \frac{286 \sin^{21}{\left(2 \right)}}{21} - \sin{\left(2 \right)} - \frac{13 \sin^{25}{\left(2 \right)}}{25} + \frac{\sin^{27}{\left(2 \right)}}{27} + \frac{78 \sin^{23}{\left(2 \right)}}{23} + \frac{13 \sin^{3}{\left(2 \right)}}{3} + \frac{715 \sin^{19}{\left(2 \right)}}{19} + \frac{286 \sin^{7}{\left(2 \right)}}{7} + \frac{572 \sin^{15}{\left(2 \right)}}{5} + 117 \sin^{11}{\left(2 \right)} + \frac{22308732928}{35102025}\right\rangle$$

                                                                17             9             21            5            25         27            3            23             7             15             19                                                                  17             9             21            5            25         27            3            23             7             15             19    
   22308732928                   13             11      1287*sin  (2)   715*sin (2)   286*sin  (2)   78*sin (2)   13*sin  (2)   sin  (2)   13*sin (2)   78*sin  (2)   286*sin (2)   572*sin  (2)   715*sin  (2)  22308732928                   13             11      1287*sin  (2)   715*sin (2)   286*sin  (2)   78*sin (2)   13*sin  (2)   sin  (2)   13*sin (2)   78*sin  (2)   286*sin (2)   572*sin  (2)   715*sin  (2) 
<- ----------- - sin(2) - 132*sin  (2) + 117*sin  (2) - ------------- - ----------- - ------------ - ---------- - ----------- + -------- + ---------- + ----------- + ----------- + ------------ + ------------, ----------- - sin(2) - 132*sin  (2) + 117*sin  (2) - ------------- - ----------- - ------------ - ---------- - ----------- + -------- + ---------- + ----------- + ----------- + ------------ + ------------>
     35102025                                                 17             9             21            5             25          27          3             23            7             5              19         35102025                                                 17             9             21            5             25          27          3             23            7             5              19

AccumBounds(-22308732928/35102025 - sin(2) - 132*sin(2)^13 + 117*sin(2)^11 - 1287*sin(2)^17/17 - 715*sin(2)^9/9 - 286*sin(2)^21/21 - 78*sin(2)^5/5 - 13*sin(2)^25/25 + sin(2)^27/27 + 13*sin(2)^3/3 + 78*sin(2)^23/23 + 286*sin(2)^7/7 + 572*sin(2)^15/5 + 715*sin(2)^19/19, 22308732928/35102025 - sin(2) - 132*sin(2)^13 + 117*sin(2)^11 - 1287*sin(2)^17/17 - 715*sin(2)^9/9 - 286*sin(2)^21/21 - 78*sin(2)^5/5 - 13*sin(2)^25/25 + sin(2)^27/27 + 13*sin(2)^3/3 + 78*sin(2)^23/23 + 286*sin(2)^7/7 + 572*sin(2)^15/5 + 715*sin(2)^19/19)

Use the examples entering the upper and lower limits of integration.

Other calculators

How to use it?

Identical expressions

Integral of cos^27x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2