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

Problem

(∫_^)(x*sin(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 the variables u and d(v) using the LIATE rule, where u is the algebraic term and d(v) is the trigonometric term.

u=x

d(v)=sin(x)*d(x)

3. Differentiate u and integrate d(v) to find d(u) and v

d(u)=d(x)

v=−cos(x)

4. Substitute these values into the Integration by Parts formula.

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

5. Simplify the expression and the signs.

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

6. Evaluate the remaining integral of cos(x)

(∫_^)(cos(x)*d(x))=sin(x)

7. Combine the terms and add the constant of integration C

−x*cos(x)+sin(x)+C

Final Answer

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

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