What is the integral of (3x^3+4x)e^{x^2} ?

The integral of (3x^3+4x)e^{x^2} is 3/2 e^{x^2}x^2+1/2 e^{x^2}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (3x^3+4x)e^{x^2}

\int (3 x^{3} + 4 x) e^{x^{2}} d x

\frac{3}{2} e^{x^{2}} x^{2} + \frac{1}{2} e^{x^{2}} + C

Solution steps

\int (3 x^{3} + 4 x) e^{x^{2}} d x

Expand

(3 x^{3} + 4 x) e^{x^{2}} : 3 e^{x^{2}} x^{3} + 4 e^{x^{2}} x

= \int 3 e^{x^{2}} x^{3} + 4 e^{x^{2}} x d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 3 e^{x^{2}} x^{3} d x + \int 4 e^{x^{2}} x d x

\int 3 e^{x^{2}} x^{3} d x = \frac{3}{2} (e^{x^{2}} x^{2} - e^{x^{2}})

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

= \frac{3}{2} (e^{x^{2}} x^{2} - e^{x^{2}}) + 2 e^{x^{2}}

Simplify

\frac{3}{2} (e^{x^{2}} x^{2} - e^{x^{2}}) + 2 e^{x^{2}} : \frac{3}{2} e^{x^{2}} x^{2} + \frac{1}{2} e^{x^{2}}

= \frac{3}{2} e^{x^{2}} x^{2} + \frac{1}{2} e^{x^{2}}

Add a constant to the solution

= \frac{3}{2} e^{x^{2}} x^{2} + \frac{1}{2} e^{x^{2}} + C

\frac{d}{d x} (\frac{1}{10} x - \frac{1}{4})

\int 3 e^{x^{2}} d x

2 - 2 cos^{2} (x)

\frac{\partial}{\partial x} ((x + 3)^{2})

n = 0 \sum \infty sin (n)

What is the integral of (3x^3+4x)e^{x^2} ?
The integral of (3x^3+4x)e^{x^2} is 3/2 e^{x^2}x^2+1/2 e^{x^2}+C