What is the integral of 2^{2x+1} ?

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

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integral of 2^{2x+1}

\int 2^{2 x + 1} d x

\frac{4 ^{x}}{ln ( 2 )} + C

Solution steps

\int 2^{2 x + 1} d x

Apply u-substitution

= \int 2^{u - 1} d u

Simplify

2^{u - 1} : 2^{u} \cdot 2^{- 1}

= \int 2^{u} \cdot 2^{- 1} d u

Take the constant out:

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

= 2^{- 1} \cdot \int 2^{u} d u

Apply exponent rule:

a^{- 1} = \frac{1}{a}

= \frac{1}{2} \cdot \int 2^{u} d u

\int a^{x} d x = \frac{a ^{x}}{ln a}

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

Substitute back

u = 2 x + 1

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

Simplify

\frac{1}{2} \cdot \frac{2 ^{2 x + 1}}{ln ( 2 )} : \frac{4 ^{x}}{ln ( 2 )}

= \frac{4 ^{x}}{ln ( 2 )}

Add a constant to the solution

= \frac{4 ^{x}}{ln ( 2 )} + C