Evaluate the Integral integral of xsin(x)^3 with respect to x

Problem

(∫_^)(x*sin^3(x)*d(x))

Solution

1. Use a trigonometric identity to rewrite sin^3(x) as sin(x)*sin^2(x) and then substitute sin^2(x)=1−cos^2(x)

(∫_^)(x*sin(x)*(1−cos^2(x))*d(x))

2. Distribute the terms inside the integral to separate it into two parts.

(∫_^)(x*sin(x)*d(x))−(∫_^)(x*sin(x)*cos^2(x)*d(x))

3. Apply integration by parts to the first integral (∫_^)(x*sin(x)*d(x)) using u=x and d(v)=sin(x)*d(x)

−x*cos(x)−(∫_^)(−cos(x)*d(x))

−x*cos(x)+sin(x)

4. Apply integration by parts to the second integral (∫_^)(x*(sin(x)*cos^2(x))*d(x)) using u=x and d(v)=sin(x)*cos^2(x)*d(x)

(∫_^)(sin(x)*cos^2(x)*d(x))=−cos^3(x)/3

x*(−cos^3(x)/3)−(∫_^)(−cos^3(x)/3*d(x))

−(x*cos^3(x))/3+1/3*(∫_^)(cos^3(x)*d(x))

5. Evaluate the integral of cos^3(x) by writing it as cos(x)*(1−sin^2(x))

(∫_^)(cos(x)*(1−sin^2(x))*d(x))=sin(x)−sin^3(x)/3

6. Combine all terms from the previous steps, ensuring the signs are handled correctly.

−x*cos(x)+sin(x)−(−(x*cos^3(x))/3+1/3*(sin(x)−sin^3(x)/3))

−x*cos(x)+sin(x)+(x*cos^3(x))/3−sin(x)/3+sin^3(x)/9

7. Simplify the expression by combining the sin(x) terms.

(x*cos^3(x))/3−x*cos(x)+(2*sin(x))/3+sin^3(x)/9+C

Final Answer

(∫_^)(x*sin^3(x)*d(x))=(x*cos^3(x))/3−x*cos(x)+(2*sin(x))/3+sin^3(x)/9+C

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