# How do you integrate #int x^2e^x dx # using integration by parts?

The integration by parts formula say

This last integral is tabled, so

By signing up, you agree to our Terms of Service and Privacy Policy

To integrate ( \int x^2 e^x , dx ) using integration by parts, we use the formula ( \int u , dv = uv - \int v , du ). Let's choose ( u = x^2 ) and ( dv = e^x , dx ). Then, ( du = 2x , dx ) and ( v = e^x ).

[ \begin{aligned} \int x^2 e^x , dx &= x^2 e^x - \int 2x e^x , dx \ &= x^2 e^x - 2 \int x e^x , dx \end{aligned} ]

Now, we apply integration by parts to ( \int x e^x , dx ) with ( u = x ) and ( dv = e^x , dx ). Then, ( du = dx ) and ( v = e^x ).

[ \begin{aligned} \int x^2 e^x , dx &= x^2 e^x - 2 (x e^x - \int e^x , dx) \ &= x^2 e^x - 2 (x e^x - e^x) \ &= x^2 e^x - 2x e^x + 2e^x + C \end{aligned} ]

Therefore, ( \int x^2 e^x , dx = x^2 e^x - 2x e^x + 2e^x + C ), where ( C ) is the constant of integration.

By signing up, you agree to our Terms of Service and Privacy Policy

- How do you integrate #(2x) / (4x^2 + 12x + 9)# using partial fractions?
- How do you find #int (5x^2+3x-2)/(x^3+2x^2) dx# using partial fractions?
- How do you integrate #int 1/sqrt(4x^2-12x+4) # using trigonometric substitution?
- How do you find the integral of #e^x *sin x#?
- How do you integrate #int 1/sqrt(e^(2x)-2e^x+10)dx# using trigonometric substitution?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7