What is the integral of 2^x(1+2^x)^4 ?

The integral of 2^x(1+2^x)^4 is ((1+2^x)^5)/(5ln(2))+C

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integral of 2^x(1+2^x)^4

\int 2^{x} (1 + 2^{x})^{4} d x

\frac{( 1 + 2 ^{x} ) ^{5}}{5 ln ( 2 )} + C

Solution steps

\int 2^{x} (1 + 2^{x})^{4} d x

Apply u-substitution

: \int \frac{u ^{4}}{ln ( 2 )} d u

= \int \frac{u ^{4}}{ln ( 2 )} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{ln ( 2 )} \cdot \int u^{4} d u

Apply the Power Rule

: \frac{u ^{5}}{5}

= \frac{1}{ln ( 2 )} \cdot \frac{u ^{5}}{5}

Substitute back

u = 1 + 2^{x}

= \frac{1}{ln ( 2 )} \cdot \frac{( 1 + 2 ^{x} ) ^{5}}{5}

Simplify

\frac{1}{ln ( 2 )} \cdot \frac{( 1 + 2 ^{x} ) ^{5}}{5} : \frac{( 1 + 2 ^{x} ) ^{5}}{5 ln ( 2 )}

= \frac{( 1 + 2 ^{x} ) ^{5}}{5 ln ( 2 )}

Add a constant to the solution

= \frac{( 1 + 2 ^{x} ) ^{5}}{5 ln ( 2 )} + C

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What is the integral of 2^x(1+2^x)^4 ?
The integral of 2^x(1+2^x)^4 is ((1+2^x)^5)/(5ln(2))+C