Mister Exam

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Integral of d{x}:
Integral of 2/x^4
Integral of sin(5x)
Integral of 1/xdx
Integral of e^(3*x)/3
Identical expressions
Sin^ three *x*cos*x*d*t
Sin cubed multiply by x multiply by co sinus of e of multiply by x multiply by d multiply by t
Sin to the power of three multiply by x multiply by co sinus of e of multiply by x multiply by d multiply by t
Sin3*x*cos*x*d*t
Sin³*x*cos*x*d*t
Sin to the power of 3*x*cos*x*d*t
Sin^3xcosxdt
Sin3xcosxdt
Sin^3*x*cos*x*d*tdx

Integral of Sin^3xcosxd*t dx

The solution

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  1                      
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 |  sin (x)*cos(x)*d*t dx
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0

$$\int\limits_{0}^{1} t d \sin^{3}{\left(x \right)} \cos{\left(x \right)}\, dx$$

Integral(((sin(x)^3*cos(x))*d)*t, (x, 0, 1))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int t d \sin^{3}{\left(x \right)} \cos{\left(x \right)}\, dx = t \int d \sin^{3}{\left(x \right)} \cos{\left(x \right)}\, dx$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int d \sin^{3}{\left(x \right)} \cos{\left(x \right)}\, dx = d \int \sin^{3}{\left(x \right)} \cos{\left(x \right)}\, dx$
  1. There are multiple ways to do this integral.
    Method #1
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{3}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{3}\, du = \frac{u^{4}}{4}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{4}{\left(x \right)}}{4}$
    Method #2
    1. Rewrite the integrand:
      
      $\sin^{3}{\left(x \right)} \cos{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}$
    2. Let $u = - \cos^{2}{\left(x \right)}$ .
      
      Then let $du = 2 \sin{\left(x \right)} \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int \left(\frac{u}{2} + \frac{1}{2}\right)\, du$
      
      Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{u}{2}\, du = \frac{\int u\, du}{2}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $\frac{u^{2}}{4}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, du = \frac{u}{2}$
      
      The result is: $\frac{u^{2}}{4} + \frac{u}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{\cos^{4}{\left(x \right)}}{4} - \frac{\cos^{2}{\left(x \right)}}{2}$
  So, the result is: $\frac{d \sin^{4}{\left(x \right)}}{4}$
So, the result is: $\frac{d t \sin^{4}{\left(x \right)}}{4}$
Add the constant of integration:

$\frac{d t \sin^{4}{\left(x \right)}}{4}+ \mathrm{constant}$

The answer is:

$\frac{d t \sin^{4}{\left(x \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                       
 |                                    4   
 |    3                        d*t*sin (x)
 | sin (x)*cos(x)*d*t dx = C + -----------
 |                                  4     
/

$$\int t d \sin^{3}{\left(x \right)} \cos{\left(x \right)}\, dx = C + \frac{d t \sin^{4}{\left(x \right)}}{4}$$

Simplify

The answer [src]

       4   
d*t*sin (1)
-----------
     4

$$\frac{d t \sin^{4}{\left(1 \right)}}{4}$$

       4   
d*t*sin (1)
-----------
     4

$$\frac{d t \sin^{4}{\left(1 \right)}}{4}$$

d*t*sin(1)^4/4

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

Other calculators

How to use it?

Identical expressions

Integral of Sin^3xcosxd*t dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Other calculators

How to use it?

Identical expressions

Integral of Sin^3*x*cos*x*d*t dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Integral of Sin^3xcosxd*t dx