What is the integral of x*4^x ?

The integral of x*4^x is (2^{2x-1}x)/(ln(2))-1/(ln^2(2))*2^{2x-2}+C

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integral of x*4^x

\int x \cdot 4^{x} d x

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

Solution steps

\int x \cdot 4^{x} d x

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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What is the integral of x*4^x ?
The integral of x*4^x is (2^{2x-1}x)/(ln(2))-1/(ln^2(2))*2^{2x-2}+C