What is the integral from-2 to 2 of (x+2)2^x ?

The integral from-2 to 2 of (x+2)2^x is (64ln(2)-15)/(4ln^2(2))

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integral from-2 to 2 of (x+2)2^x

\int_{- 2}^{2} (x + 2) 2^{x} d x

\frac{64 ln ( 2 ) - 15}{4 ln ^{2} ( 2 )}

Decimal

15.27798 \dots

Solution steps

\int_{- 2}^{2} (x + 2) \cdot 2^{x} d x

Expand

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

= \int_{- 2}^{2} 2^{x} x + 2^{1 + 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}^{2} 2^{x} x d x + \int_{- 2}^{2} 2^{1 + x} d x

\int_{- 2}^{2} 2^{x} x d x = \frac{34 ln ( 2 ) - 15}{4 ln ^{2} ( 2 )}

\int_{- 2}^{2} 2^{1 + x} d x = \frac{15}{2 ln ( 2 )}

= \frac{34 ln ( 2 ) - 15}{4 ln ^{2} ( 2 )} + \frac{15}{2 ln ( 2 )}

Simplify

\frac{34 ln ( 2 ) - 15}{4 ln ^{2} ( 2 )} + \frac{15}{2 ln ( 2 )} : \frac{64 ln ( 2 ) - 15}{4 ln ^{2} ( 2 )}

= \frac{64 ln ( 2 ) - 15}{4 ln ^{2} ( 2 )}

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What is the integral from-2 to 2 of (x+2)2^x ?
The integral from-2 to 2 of (x+2)2^x is (64ln(2)-15)/(4ln^2(2))