What is the integral of (x^3+x^2+x-1)e^{2x} ?

The integral of (x^3+x^2+x-1)e^{2x} is 1/2 e^{2x}x^3-1/4 e^{2x}x^2+3/4 e^{2x}x-7/8 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 (x^3+x^2+x-1)e^{2x}

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

\frac{1}{2} e^{2 x} x^{3} - \frac{1}{4} e^{2 x} x^{2} + \frac{3}{4} e^{2 x} x - \frac{7}{8} e^{2 x} + C

Solution steps

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

Expand

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

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

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

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

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

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

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

Simplify

\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})) + \frac{1}{2} e^{2 x} x^{2} - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x}) + \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x}) - \frac{1}{2} e^{2 x} : \frac{1}{2} e^{2 x} x^{3} - \frac{1}{4} e^{2 x} x^{2} + \frac{3}{4} e^{2 x} x - \frac{7}{8} e^{2 x}

= \frac{1}{2} e^{2 x} x^{3} - \frac{1}{4} e^{2 x} x^{2} + \frac{3}{4} e^{2 x} x - \frac{7}{8} e^{2 x}

Add a constant to the solution

= \frac{1}{2} e^{2 x} x^{3} - \frac{1}{4} e^{2 x} x^{2} + \frac{3}{4} e^{2 x} x - \frac{7}{8} e^{2 x} + C

\int \frac{6 + sin ( x )}{cos ( x )} d x

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

\int e^{2 x} \cdot cos (6 x) d x

\int (x^{2} - 2) cos (2 x) d x

\int (sin (x) sin (3 x)) d x

What is the integral of (x^3+x^2+x-1)e^{2x} ?
The integral of (x^3+x^2+x-1)e^{2x} is 1/2 e^{2x}x^3-1/4 e^{2x}x^2+3/4 e^{2x}x-7/8 e^{2x}+C