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Popular Calculus >

integral of (x+1)^2*3^{-x}

Solution

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

Solution

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

Solution steps

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

Expand

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

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

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

Graph

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Frequently Asked Questions (FAQ)

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