What is the integral of e^{2x}*(2x^2+8x+3) ?

The integral of e^{2x}*(2x^2+8x+3) is e^{2x}x^2+3e^{2x}x+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of e^{2x}*(2x^2+8x+3)

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

e^{2 x} x^{2} + 3 e^{2 x} x + C

Solution steps

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

Expand

e^{2 x} (2 x^{2} + 8 x + 3) : 2 e^{2 x} x^{2} + 8 e^{2 x} x + 3 e^{2 x}

= \int 2 e^{2 x} x^{2} + 8 e^{2 x} x + 3 e^{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 2 e^{2 x} x^{2} d x + \int 8 e^{2 x} x d x + \int 3 e^{2 x} d x

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

\int 8 e^{2 x} x d x = 2 (2 e^{2 x} x - e^{2 x})

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

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

Simplify

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

= e^{2 x} x^{2} + 3 e^{2 x} x

Add a constant to the solution

= e^{2 x} x^{2} + 3 e^{2 x} x + C

\int (12 x^{3} + 6 x - 2) d x

\int \frac{3}{4 x ^{2}} d x

\int \frac{e ^{8 x}}{e ^{8 x} + 3} d x

\int x^{2} + 2 x + 5 d x

\int \frac{1}{( a ^{2} + z ^{2} ) ^{\frac{3}{2}}} d z

What is the integral of e^{2x}*(2x^2+8x+3) ?
The integral of e^{2x}*(2x^2+8x+3) is e^{2x}x^2+3e^{2x}x+C