What is the integral of (5x^3+6x)e^{x^2} ?

The integral of (5x^3+6x)e^{x^2} is 5/2 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 (5x^3+6x)e^{x^2}

\int (5 x^{3} + 6 x) e^{x^{2}} d x

\frac{5}{2} e^{x^{2}} x^{2} + \frac{1}{2} e^{x^{2}} + C

Solution steps

\int (5 x^{3} + 6 x) e^{x^{2}} d x

Expand

(5 x^{3} + 6 x) e^{x^{2}} : 5 e^{x^{2}} x^{3} + 6 e^{x^{2}} x

= \int 5 e^{x^{2}} x^{3} + 6 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 5 e^{x^{2}} x^{3} d x + \int 6 e^{x^{2}} x d x

\int 5 e^{x^{2}} x^{3} d x = \frac{5}{2} (e^{x^{2}} x^{2} - e^{x^{2}})

\int 6 e^{x^{2}} x d x = 3 e^{x^{2}}

= \frac{5}{2} (e^{x^{2}} x^{2} - e^{x^{2}}) + 3 e^{x^{2}}

Simplify

\frac{5}{2} (e^{x^{2}} x^{2} - e^{x^{2}}) + 3 e^{x^{2}} : \frac{5}{2} e^{x^{2}} x^{2} + \frac{1}{2} e^{x^{2}}

= \frac{5}{2} e^{x^{2}} x^{2} + \frac{1}{2} e^{x^{2}}

Add a constant to the solution

= \frac{5}{2} e^{x^{2}} x^{2} + \frac{1}{2} e^{x^{2}} + C

\int \frac{1}{9 y ^{2} + 4} d y

\int (1 - x) d x

e^{- 4 t} \times sin (2 t)

\frac{10}{3} x^{3} - 5 x^{2} - 11 x + 6

\int \frac{x}{100 x ^{2} - 1} d x

What is the integral of (5x^3+6x)e^{x^2} ?
The integral of (5x^3+6x)e^{x^2} is 5/2 e^{x^2}x^2+1/2 e^{x^2}+C