What is the integral of 2x^3(e^x)^2 ?

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

integral

2 x^{3} (e^{x})^{2}

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

Solution steps

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

Take the constant out:

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

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

(e^{x})^{2} = e^{2 x}

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

Apply Integration By Parts

: \frac{1}{2} e^{2 x} x^{3} - \int \frac{3}{2} e^{2 x} x^{2} d x

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

\int \frac{3}{2} e^{2 x} x^{2} d x = \frac{3}{2} (\frac{1}{2} e^{2 x} x^{2} - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x}))

= 2 (\frac{1}{2} e^{2 x} x^{3} - \frac{3}{2} (\frac{1}{2} e^{2 x} x^{2} - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x})))

Simplify

2 (\frac{1}{2} e^{2 x} x^{3} - \frac{3}{2} (\frac{1}{2} e^{2 x} x^{2} - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x}))) : e^{2 x} x^{3} - \frac{3}{2} e^{2 x} x^{2} + \frac{3}{2} e^{2 x} x - \frac{3}{4} e^{2 x}

= e^{2 x} x^{3} - \frac{3}{2} e^{2 x} x^{2} + \frac{3}{2} e^{2 x} x - \frac{3}{4} e^{2 x}

Add a constant to the solution

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

2 x^{3} \cdot e^{- x}

\frac{y}{x ^{2} + y ^{2}}

2 + x

\frac{1}{( x ^{\frac{1}{2}} + 1 )}

(x^{2} - x)^{4}

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