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

The integral of x(e^x+e^{x^2}) is e^xx-e^x+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 x(e^x+e^{x^2})

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

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

Solution steps

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

Expand

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

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

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

\int e^{x^{2}} x d x = \frac{1}{2} e^{x^{2}}

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

Add a constant to the solution

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

2 x^{2} + 11, 0, - 2, 2

f (x) = x^{3} - 2 x + 2, at x = 4

\frac{d}{d x} (ln (x + ln (x)))

\frac{d}{d x} (1 - e^{- \frac{x}{2}})

\frac{1}{x - 5}

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