Evaluate the Integral integral of 2xcos(x) with respect to x

Problem

(∫_^)(2*x*cos(x)*d(x))

Solution

1. Identify the integration method. Since the integrand is a product of an algebraic function 2*x and a trigonometric function cos(x) use integration by parts.

2. Choose the parts for the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=2*x and d(v)=cos(x)*d(x)

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

d(u)=2*d(x)

v=sin(x)

4. Apply the integration by parts formula.

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

5. Simplify the expression and prepare for the second integral.

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

6. Evaluate the remaining integral. The integral of sin(x) is −cos(x)

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

7. Simplify the signs and add the constant of integration C

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

Final Answer

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

Want more problems? Check here!