Evaluate the Integral integral of xcos(x) with respect to x

Problem

(∫_^)(x*cos(x)*d(x))

Solution

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

2. Choose the variables u and d(v) using the LIATE rule, setting u=x and d(v)=cos(x)*d(x)

3. Differentiate u to find d(u)=d(x)

4. Integrate d(v) to find v=sin(x)

5. Substitute these components into the integration by parts formula.

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

6. Evaluate the remaining integral (∫_^)(sin(x)*d(x))=−cos(x)

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

7. Simplify the signs to reach the final expression.

Final Answer

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

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