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

The integral of 3^{sqrt(2x+1)} is (3^{sqrt(2x+1)}sqrt(2x+1))/(ln(3))-(3^{sqrt(2x+1)})/(ln^2(3))+C

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

\int 3^{2 x + 1} d x

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

Solution steps

\int 3^{2 x + 1} d x

Apply u-substitution

: \int 3^{u} u d u

= \int 3^{u} u d u

Apply Integration By Parts

: \frac{3 ^{u} u}{ln ( 3 )} - \int \frac{3 ^{u}}{ln ( 3 )} d u

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

\int \frac{3 ^{u}}{ln ( 3 )} d u = \frac{3 ^{u}}{ln ^{2} ( 3 )}

= \frac{3 ^{u} u}{ln ( 3 )} - \frac{3 ^{u}}{ln ^{2} ( 3 )}

Substitute back

u = 2 x + 1

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

Add a constant to the solution

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

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What is the integral of 3^{sqrt(2x+1)} ?
The integral of 3^{sqrt(2x+1)} is (3^{sqrt(2x+1)}sqrt(2x+1))/(ln(3))-(3^{sqrt(2x+1)})/(ln^2(3))+C