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^3-5/4 e^{2x}x^2+3/4 e^{2x}x+1/8 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{5}{4} e^{2 x} x^{2} + \frac{3}{4} e^{2 x} x + \frac{1}{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{5}{4} e^{2 x} x^{2} + \frac{3}{4} e^{2 x} x + \frac{1}{8} e^{2 x}

= \frac{1}{2} e^{2 x} x^{3} - \frac{5}{4} e^{2 x} x^{2} + \frac{3}{4} e^{2 x} x + \frac{1}{8} e^{2 x}

Add a constant to the solution

= \frac{1}{2} e^{2 x} x^{3} - \frac{5}{4} e^{2 x} x^{2} + \frac{3}{4} e^{2 x} x + \frac{1}{8} e^{2 x} + C

\int \frac{1}{4 x ^{2} + 8 x + 19} d x

\int ln^{2} (\frac{3 a}{x}) d x

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

\int 20 x^{3} (5 x^{4} - 3)^{4} d x

\int (- sec^{2} (x) + e^{2 x + 1}) 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^3-5/4 e^{2x}x^2+3/4 e^{2x}x+1/8 e^{2x}+C