Evaluate the Integral integral of x*cos(x) with respect to x

Problem

(∫_^)(x*cos(x)*d(x))

Solution

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

2. Choose the variables for substitution using the LIATE rule, 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. Substitute these components into the integration by parts formula.

(∫_^)(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 signs to reach the final expression.

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

Final Answer

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

Want more problems? Check here!