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

The integral of 2x*e^{2x} is e^{2x}x-1/2 e^{2x}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 2x*e^{2x}

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

e^{2 x} x - \frac{1}{2} e^{2 x} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{e ^{u} u}{4} d u

= 2 \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

= 2 \cdot \frac{1}{4} \cdot \int e^{u} u d u

Apply Integration By Parts

: e^{u} u - \int e^{u} d u

= 2 \cdot \frac{1}{4} (e^{u} u - \int e^{u} d u)

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

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

Substitute back

u = 2 x

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

Simplify

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

= e^{2 x} x - \frac{1}{2} e^{2 x}

Add a constant to the solution

= e^{2 x} x - \frac{1}{2} e^{2 x} + C

\int (1 + \frac{1}{u})^{- 3} (\frac{1}{u ^{2}}) d u

\frac{d}{d x} (4 x + 8 x^{\frac{3}{8}})

\frac{d}{d x} \frac{( x ^{2} - 4 ) ^{3}}{36}

x \to 1 + lim (\frac{1}{x ^{3}})

(- 5, - 1), (0, 3)

What is the integral of 2x*e^{2x} ?
The integral of 2x*e^{2x} is e^{2x}x-1/2 e^{2x}+C