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Popular >

integral of (x^2+3x)2^x

Solution

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

Solution

\frac{2 ^{x} x ^{2}}{ln ( 2 )} - \frac{1}{ln ( 2 )} (\frac{2 ^{x + 1} x}{ln ( 2 )} - \frac{1}{ln ^{2} ( 2 )} \cdot 2^{1 + x}) + 3 (\frac{2 ^{x} x}{ln ( 2 )} - \frac{2 ^{x}}{ln ^{2} ( 2 )}) + C

Solution steps

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

Expand

(x^{2} + 3 x) \cdot 2^{x} : 2^{x} x^{2} + 3 \cdot 2^{x} x

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

\int 2^{x} x^{2} d x = \frac{2 ^{x} x ^{2}}{ln ( 2 )} - \frac{1}{ln ( 2 )} (\frac{2 ^{x + 1} x}{ln ( 2 )} - \frac{1}{ln ^{2} ( 2 )} \cdot 2^{1 + x})

\int 3 \cdot 2^{x} x d x = 3 (\frac{2 ^{x} x}{ln ( 2 )} - \frac{2 ^{x}}{ln ^{2} ( 2 )})

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

Add a constant to the solution

= \frac{2 ^{x} x ^{2}}{ln ( 2 )} - \frac{1}{ln ( 2 )} (\frac{2 ^{x + 1} x}{ln ( 2 )} - \frac{1}{ln ^{2} ( 2 )} \cdot 2^{1 + x}) + 3 (\frac{2 ^{x} x}{ln ( 2 )} - \frac{2 ^{x}}{ln ^{2} ( 2 )}) + C

Popular Examples

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\int \frac{x ^{3} + 4 x - 13}{x ^{2} + 4} d x

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\int 13 d x

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x \to 7 lim (5 x^{- 1})

integral of x/(x^2-2x-3)

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

Frequently Asked Questions (FAQ)

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