Question: "What is the integral of 2xex?"

Bradyn Cherry

Answered question

2023-01-14

What is the integral of

2 x e^{x}

possetzvjm

Beginner2023-01-15Added 7 answers

Need find:

\int 2 x e^{x} d x

By constant rule

\int 2 x e^{x} d x = 2 \int x e^{x} d x

Let

I = \int x e^{x} d x

\int 2 x e^{x} d x = 2 I

The parts rule allows us to integrate.

\int u d v = u v - \int v d u

Here,

u = x \to d u = d x

d v = e^{x} d x \to v = e^{x}

Substituting

\int x e^{x} d x = x e^{x} - \int e^{x} d x

\int e^{x} d x = e^{x}

\int x e^{x} d x = x e^{x} - e^{x}

I = x e^{x} - e^{x}

\int 2 x e^{x} d x = 2 (x e^{x} - e^{x}) + C

Audrina Graham

Beginner2023-01-16Added 1 answers

Given:

\int (2 x e^{x}) d x

Integration by parts can be used to evaluate this integral.
Let

u = 2 x \Rightarrow \frac{d u}{d x} = 2

and

\frac{d v}{d x} = e^{x} \Rightarrow v = e^{x}

\Rightarrow \int (2 x e^{x}) d x = 2 x e^{x} - \int (2 e^{x}) d x

∴ \int (2 x e^{x}) d x = 2 x e^{x} - 2 e^{x} + C

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