Evaluate the Integral integral of x^2cos(x) with respect to x

Problem

(∫_^)(x2*cos(x)*d(x))

Solution

1. Identify the integration method as integration by parts, which uses the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Choose u=x2 and d(v)=cos(x)*d(x) which implies d(u)=2*x*d(x) and v=sin(x)

3. Apply the integration by parts formula for the first time.

(∫_^)(x2*cos(x)*d(x))=x2*sin(x)−(∫_^)(2*x*sin(x)*d(x))

4. Apply integration by parts again to the remaining integral (∫_^)(2*x*sin(x)*d(x)) by choosing u=2*x and d(v)=sin(x)*d(x) which implies d(u)=2*d(x) and v=−cos(x)

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

5. Evaluate the integral of the constant multiple of the cosine function.

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

6. Substitute this result back into the original equation and include the constant of integration C

(∫_^)(x2*cos(x)*d(x))=x2*sin(x)−(−2*x*cos(x)+2*sin(x))+C

7. Simplify the signs to reach the final expression.

(∫_^)(x2*cos(x)*d(x))=x2*sin(x)+2*x*cos(x)−2*sin(x)+C

Final Answer

(∫_^)(x2*cos(x)*d(x))=x2*sin(x)+2*x*cos(x)−2*sin(x)+C

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