Evaluate the Integral

Problem

(∫_0^π)(x*sin(x)*d(x))

Solution

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

2. Assign the variables for integration by parts by letting u=x and d(v)=sin(x)*d(x)

3. Differentiate u to find d(u)=d(x) and integrate d(v) to find v=−cos(x)

4. Apply the formula for integration by parts to the indefinite integral.

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

5. Simplify the integral of the second term.

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

6. Evaluate the remaining integral to find the general antiderivative.

(∫_^)(x*sin(x)*d(x))=−x*cos(x)+sin(x)

7. Apply the limits of integration from 0 to π using the Fundamental Theorem of Calculus.

[−x*cos(x)+sin(x)]π0

8. Substitute the upper limit π and the lower limit 0 into the expression.

(−π*cos(π)+sin(π))−(−0*cos(0)+sin(0))

9. Simplify the trigonometric values where cos(π)=−1 sin(π)=0 cos(0)=1 and sin(0)=0

(−π*(−1)+0)−(0+0)

10. Calculate the final numerical result.

π

Final Answer

(∫_0^π)(x*sin(x)*d(x))=π

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