What is the integral of 4x*e^{2x} ?

The integral of 4x*e^{2x} is 2e^{2x}x-e^{2x}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of 4x*e^{2x}

\int 4 x \cdot e^{2 x} d x

2 e^{2 x} x - e^{2 x} + C

Solution steps

\int 4 x e^{2 x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int x e^{2 x} d x

Apply u-substitution

= 4 \cdot \int \frac{e ^{u} u}{4} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \frac{1}{4} \cdot \int e^{u} u d u

Apply Integration By Parts

= 4 \cdot \frac{1}{4} (e^{u} u - \int e^{u} d u)

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

= 4 \cdot \frac{1}{4} (e^{u} u - e^{u})

Substitute back

u = 2 x

= 4 \cdot \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x})

Simplify

4 \cdot \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x}) : 2 e^{2 x} x - e^{2 x}

= 2 e^{2 x} x - e^{2 x}

Add a constant to the solution

= 2 e^{2 x} x - e^{2 x} + C

What is the integral of 4x*e^{2x} ?
The integral of 4x*e^{2x} is 2e^{2x}x-e^{2x}+C