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

The integral of e^{x/3}(x^2+2x) is 3e^{x/3}x^2-4(3e^{x/3}x-9e^{x/3})+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

3 e^{\frac{x}{3}} x^{2} - 4 (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}}) + C

Solution steps

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

Expand

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

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

\int e^{\frac{x}{3}} x^{2} d x = 3 e^{\frac{x}{3}} x^{2} - 6 (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}})

\int 2 e^{\frac{x}{3}} x d x = 2 (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}})

= 3 e^{\frac{x}{3}} x^{2} - 6 (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}}) + 2 (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}})

Simplify

= 3 e^{\frac{x}{3}} x^{2} - 4 (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}})

Add a constant to the solution

= 3 e^{\frac{x}{3}} x^{2} - 4 (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}}) + C

\int 24 x (2 x^{2} + 5)^{2} d x

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

\int \frac{1}{x ^{3} + 6 x ^{2} + 11 x + 6} d x

\int \frac{t ^{2} - 1}{t} d t

\int (8 x^{3} + 3 3 x^{2}) d x

What is the integral of e^{x/3}(x^2+2x) ?
The integral of e^{x/3}(x^2+2x) is 3e^{x/3}x^2-4(3e^{x/3}x-9e^{x/3})+C