Find the Integral xcos(x)

Problem

(∫_^)(x*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 the variables for substitution by letting u=x and d(v)=cos(x)*d(x)

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

4. Apply the formula for integration by parts by substituting the values of u v d(u) and d(v)

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

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

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

6. Simplify the expression by distributing the negative sign.

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

Final Answer

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

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