Question

Exercise is mixed —some require integration by parts, while others can be integrated by using techniques discussed in the chapter on Integration.

∫^2_1 (1-x^2)e^2x dx.

Answer 1

`-42.79587`.

Explanation:

We have been given the indefinite integral
`\int _1^2\:\left(1-x^2\right)e^(2x)\:dx`. We are asked to find the integral using integration by parts.

We will use Integration by parts formula to solve our given problem.

`\int\ vdv=uv-\int\ vdu`

Let
`u=1-x^2` and
`v'=e^(2x)`.

Now, we need to find du and v using these values as shown below:

`(du)/(dx)=(d)/(dx)(1-x^2)`

`(du)/(dx)=-2x`

`du=-2xdx`

`v'=e^(2x)`

`v=(1)/(2)e^(2x)`

Upon substituting these values in integration by parts formula, we will get:

`\int _1^2\:\left(1-x^2\right)e^(2x)\:dx=(1-x^2)((e^(2x))/(2))-\int _1^2((e^(2x))/(2))(-2x)dx`

`\int _1^2\:\left(1-x^2\right)e^(2x)\:dx=(1-x^2)((e^(2x))/(2))-\int _1^2 -xe^(2x)dx`

`\int _1^2\:\left(1-x^2\right)e^(2x)\:dx=(1-x^2)((e^(2x))/(2))+(1)/(4)(e^(2x)(2x)-e^(2x))`

Let us compute the boundaries.

`(1-2^2)((e^(2*2))/(2))+(1)/(4)(e^(2*2)(2*2)-e^(2*2))=-(3e^4)/(4)=-40.94861`

`(1-1^2)((e^(2*1))/(2))+(1)/(4)(e^(2*1)(2*1)-e^(2*1))=(e^2)/(4)=1.84726`

`-40.94861-1.84726=-42.79587`

Therefore, the value pf the given definite integral would be
`-42.79587`.