Evaluate the integral, using the substitution [tex]$u=1+\cos ^{12}(x)$[/tex].
[tex]$\int\left(1+\cos ^{12}(x)\right)^9 \cos ^{11}(x) \sin (x) d x$[/tex]
1. Using [tex]$u=1+\cos ^{12}(x)$[/tex], what is [tex]$d u$[/tex]?
[tex]$d u=$[/tex] [ ] [tex]$d x$[/tex]
2. What is the integral after substitution (i.e., in terms of [tex]$u$[/tex] ?)
[tex]$\int$[/tex] [ ] [tex]$d u$[/tex]
3. What do you get by integrating the previous answer?
[ ] Give in terms of [tex]$u$[/tex]. Don't forget [tex]$+C$[/tex]
4. What is the final answer?
[ ] Answer in terms of original variable.
Answer (2)
To evaluate the integral, we substitute u = 1 + cos 12 ( x ) and find d u = − 12 cos 11 ( x ) sin ( x ) d x . This transforms the integral into a simpler form: − 12 1 ∫ u 9 d u , which we compute to obtain the final result. Substituting back for u gives the answer: − 120 ( 1 + c o s 12 ( x ) ) 10 + C .
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Find d u using the substitution u = 1 + cos 12 ( x ) , which gives d u = − 12 cos 11 ( x ) sin ( x ) d x .
Substitute u and d u into the integral, transforming it into − 12 1 ∫ u 9 d u .
Evaluate the integral with respect to u , resulting in − 120 u 10 + C .
Substitute back u = 1 + cos 12 ( x ) to obtain the final answer: − 120 ( 1 + cos 12 ( x ) ) 10 + C .
Explanation
Problem Analysis We are given the integral ∫ ( 1 + cos 12 ( x ) ) 9 cos 11 ( x ) sin ( x ) d x and the substitution u = 1 + cos 12 ( 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 12 ( x ) , we differentiate both sides with respect to x :
d x d u = d x d ( 1 + cos 12 ( x )) = 0 + 12 cos 11 ( x ) ⋅ ( − sin ( x )) = − 12 cos 11 ( x ) sin ( x ) So, d u = − 12 cos 11 ( x ) sin ( x ) d x .
Substituting u and du Now we need to rewrite the integral in terms of u . We have: ∫ ( 1 + cos 12 ( x ) ) 9 cos 11 ( x ) sin ( x ) d x = ∫ u 9 cos 11 ( x ) sin ( x ) d x From the expression for d u , we have cos 11 ( x ) sin ( x ) d x = − 12 1 d u . Substituting this into the integral, we get: ∫ u 9 cos 11 ( x ) sin ( x ) d x = ∫ u 9 ( − 12 1 ) d u = − 12 1 ∫ u 9 d u
Integrating with respect to u Next, we evaluate the integral with respect to u :
− 12 1 ∫ u 9 d u = − 12 1 ⋅ 10 u 10 + C = − 120 u 10 + C
Substituting back for x Finally, we substitute back u = 1 + cos 12 ( x ) to get the answer in terms of x :
− 120 u 10 + C = − 120 ( 1 + c o s 12 ( x ) ) 10 + C
Final Answer Therefore, the final answer is: − 120 ( 1 + c o s 12 ( x ) ) 10 + C
Examples
Imagine you're calculating the total energy absorbed by a solar panel over a day. The intensity of sunlight varies with time, often described by trigonometric functions. If the panel's absorption rate depends on a power of the sunlight intensity, you might encounter integrals similar to this one. Using substitution simplifies the calculation, allowing you to determine the total energy absorbed and optimize the panel's design.
