Question

Is it true that the integral LaTeX: \int x^2e^{2x}dx ∫ x 2 e 2 x d x can be evaluated using integration by parts? If so, state that it is and explain why. If not, state that it is not and provide a counterexample.

Answer 1

Yes the integral can be evaluated by integration by parts as solved below.

Explanation:

`\int x^(2)e^(2x)dx`

Taking algebraic function as first function and exponential function as second function we have

`\int x^(2)e^(2x)dx=x^(2)\int e^(2x)dx-\int (x^(2))'\int e^(2x)dx\\\\=x^(2)(e^(2x))/(2)-\int 2x* (e^(2x))/(2)dx\\\\(x^(2)e^(2x))/(2)-\int xe^(2x)dx\\\\Now\\\\\int xe^(2x)dx=x\int e^(2x)dx-\int 1\cdot \int e^(2x)dx\\\\=(xe^(2x))/(2)-\int (e^(2x))/(2)dx\\\\(xe^(2x))/(2)-(e^(2x))/(4)\\\\\therefore \int x^(2)e^(2x)dx=(x^(2)e^(2x))/(2)-(xe^(2x))/(2)+(e^(2x))/(4)`