Evaluate the Integral

Problem

(∫_^)(sin4(x)*cos3(x)*d(x))

Solution

1. Use a trigonometric identity to rewrite the odd power of cosine by separating one cos(x) factor.

(∫_^)(sin4(x)*cos2(x)*cos(x)*d(x))

2. Apply the Pythagorean identity cos2(x)=1−sin2(x) to express the even power of cosine in terms of sine.

(∫_^)(sin4(x)*(1−sin2(x))*cos(x)*d(x))

3. Choose a substitution u=sin(x) which implies that d(u)=cos(x)*d(x)

(∫_^)(u4*(1−u2)*d(u))

4. Distribute the u4 term into the parentheses to prepare for integration.

(∫_^)((u4−u6)*d(u))

5. Integrate each term using the power rule (∫_^)(un*d(u))=(u(n+1))/(n+1)

(u5)/5−(u7)/7+C

6. Substitute back the original expression u=sin(x) to find the final result.

sin5(x)/5−sin7(x)/7+C

Final Answer

(∫_^)(sin4(x)*cos3(x)*d(x))=sin5(x)/5−sin7(x)/7+C

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