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Integral of cos^27x dx

Limits of integration:

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Piecewise:

The solution

You have entered [src]
 oo            
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 |     27      
 |  cos  (x) dx
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2              
$$\int\limits_{2}^{\infty} \cos^{27}{\left(x \right)}\, dx$$
Integral(cos(x)^27, (x, 2, oo))
Detail solution
  1. Rewrite the integrand:

  2. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

    2. Integrate term-by-term:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

      1. The integral of cosine is sine:

      The result is:

    Method #2

    1. Rewrite the integrand:

    2. Integrate term-by-term:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

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

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        So, the result is:

      1. The integral of cosine is sine:

      The result is:

  3. Now simplify:

  4. Add the constant of integration:


The answer is:

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