# How do you find the integral from 0 to 2 of #xe^(2x) dx#?

Make use of integration by parts

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

To find the integral of ( xe^{2x} ) from 0 to 2, you can use integration by parts. Let ( u = x ) and ( dv = e^{2x} dx ). Then, ( du = dx ) and ( v = \frac{1}{2} e^{2x} ).

Applying the integration by parts formula: [ \int u , dv = uv - \int v , du ]

We get: [ \int x e^{2x} , dx = \frac{1}{2} xe^{2x} - \frac{1}{2} \int e^{2x} , dx ]

Now, integrate ( e^{2x} ): [ \int e^{2x} , dx = \frac{1}{2} e^{2x} + C ]

Substitute this back into the original equation: [ \int x e^{2x} , dx = \frac{1}{2} xe^{2x} - \frac{1}{4} e^{2x} + C ]

Now, evaluate the definite integral from 0 to 2: [ \int_{0}^{2} x e^{2x} , dx = \left[ \frac{1}{2} xe^{2x} - \frac{1}{4} e^{2x} \right]_{0}^{2} ] [ = \left( \frac{1}{2} \cdot 2 \cdot e^{4} - \frac{1}{4} \cdot e^{4} \right) - \left( \frac{1}{2} \cdot 0 \cdot e^{0} - \frac{1}{4} \cdot e^{0} \right) ] [ = \frac{2e^{4}}{2} - \frac{e^{4}}{4} - \frac{e^{0}}{4} ] [ = e^{4} - \frac{1}{4} ]

So, the integral from 0 to 2 of ( xe^{2x} ) is ( e^{4} - \frac{1}{4} ).

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

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you integrate #int t^5/sqrt(t^2+2)# by trigonometric substitution?
- How do you integrate #f(x)=(-x^2-2x)/((x^2+2)(x+7))# using partial fractions?
- How do you use partial fraction decomposition to decompose the fraction to integrate #26/(6x^2+5x-6)#?
- How do you integrate #1/((x^3)(x-4))# using partial fractions?
- How do you evaluate the integral #int sec^2x/(1+tanx)dx#?

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