Evaluate the integral, using the substitution [tex]$u=1+\cos ^5(x)$[/tex].
[tex]$\int\left(1+\cos ^5(x)\right)^6 \cos ^4(x) \sin (x) d x$[/tex]

1. Using [tex]$u=1+\cos ^5(x)$[/tex], what is [tex]$d u$[/tex]?
[tex]$d u=\square d x$[/tex]
2. What is the integral after substitution (i.e., in terms of [tex]$u$[/tex]?)
[tex]$\int \square d u$[/tex]
3. What do you get by integrating the previous answer?
[tex]$\square$[/tex] Give in terms of [tex]$u$[/tex]. Don't forget [tex]$+C$[/tex]
4. What is the final answer?
[tex]$\square$[/tex] Answer in terms of original variable.

Using the substitution u = 1 + cos 5 ( x ) , we find d u = − 5 cos 4 ( x ) sin ( x ) d x . The integral becomes − 5 1 ​ ∫ u 6 d u , which integrates to − 35 u 7 ​ + C . Finally, substituting back gives the answer − 35 ( 1 + c o s 5 ( x ) ) 7 ​ + C .
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Answered by Anonymous | 2025-07-04

Find d u using the substitution u = 1 + cos 5 ( x ) , which gives d u = − 5 cos 4 ( x ) sin ( x ) d x .
Substitute u and d u into the integral, resulting in − 5 1 ​ ∫ u 6 d u .
Integrate with respect to u to get − 35 u 7 ​ + C .
Substitute back u = 1 + cos 5 ( x ) to obtain the final answer: − 35 ( 1 + cos 5 ( x ) ) 7 ​ + C ​ .

Explanation

Problem Analysis We are given the integral ∫ ( 1 + cos 5 ( x ) ) 6 cos 4 ( x ) sin ( x ) d x and the substitution u = 1 + cos 5 ( x ) . Our goal is to evaluate the integral using the given substitution.

Finding du First, we need to find d u in terms of d x . Given u = 1 + cos 5 ( x ) , we differentiate both sides with respect to x :
d x d u ​ = d x d ​ ( 1 + cos 5 ( x )) = 0 + 5 cos 4 ( x ) ( − sin ( x )) = − 5 cos 4 ( x ) sin ( x ) So, d u = − 5 cos 4 ( x ) sin ( x ) d x .

Substituting into the Integral Now we need to express the integral in terms of u and d u . We have: ∫ ( 1 + cos 5 ( x ) ) 6 cos 4 ( x ) sin ( x ) d x Since u = 1 + cos 5 ( x ) , we have u 6 = ( 1 + cos 5 ( x ) ) 6 . Also, d u = − 5 cos 4 ( x ) sin ( x ) d x , so cos 4 ( x ) sin ( x ) d x = − 5 1 ​ d u . Substituting these into the integral, we get: ∫ u 6 ( − 5 1 ​ ) d u = − 5 1 ​ ∫ u 6 d u

Integrating with respect to u Next, we evaluate the integral with respect to u :
− 5 1 ​ ∫ u 6 d u = − 5 1 ​ ⋅ 7 u 7 ​ + C = − 35 u 7 ​ + C

Substituting back for x Finally, we substitute back x in place of u to get the final answer in terms of the original variable x . Since u = 1 + cos 5 ( x ) , we have: − 35 u 7 ​ + C = − 35 ( 1 + c o s 5 ( x ) ) 7 ​ + C

Final Answer Therefore, the final answer is: − 35 ( 1 + c o s 5 ( x ) ) 7 ​ + C


Examples
Imagine you're designing a sound wave filter that needs to dampen certain frequencies. The integral you solved helps model the energy dissipation within the filter material as it interacts with sound waves. By understanding how different components of the sound wave (represented by trigonometric functions) contribute to the overall energy absorption, you can fine-tune the filter's design for optimal performance. This ensures that unwanted noise is minimized while preserving the clarity of desired sounds.

Answered by GinnyAnswer | 2025-07-04