Evaluate the Integral

Problem

(∫_^)(sin(ln(x))*d(x))

Solution

1. Apply integration by parts by letting u=sin(ln(x)) and d(v)=d(x)

2. Calculate the differentials d(u)=cos(ln(x))/x*d(x) and v=x

3. Substitute into the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) to get:

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

4. Simplify the integral by canceling x

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

5. Apply integration by parts again to the new integral (∫_^)(cos(ln(x))*d(x)) with u=cos(ln(x)) and d(v)=d(x)

6. Calculate the differentials d(u)=−sin(ln(x))/x*d(x) and v=x

7. Substitute into the formula for the second integral:

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

8. Simplify the second integral:

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

9. Combine the results back into the original equation:

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

10. Distribute the negative sign:

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

11. Add the integral to both sides to solve for (∫_^)(sin(ln(x))*d(x))

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

12. Divide by 2 and add the constant of integration C

Final Answer

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

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