What is the integral of e^{6x}2^{9x} ?

The integral of e^{6x}2^{9x} is 1/(3(3ln(2)+2))*512^{((2+3ln(2))x)/(3ln(2))}+C

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integral of e^{6x}2^{9x}

\int e^{6 x} 2^{9 x} d x

\frac{1}{3 ( 3 ln ( 2 ) + 2 )} \cdot 51 2^{\frac{( 2 + 3 l n ( 2 ) ) x}{3 l n ( 2 )}} + C

Solution steps

\int e^{6 x} \cdot 2^{9 x} d x

2^{9 x} = 51 2^{x}

= \int 51 2^{x} e^{6 x} d x

Apply u-substitution

: \int \frac{u ^{\frac{2}{3 l n ( 2 )}}}{9 ln ( 2 )} d u

= \int \frac{u ^{\frac{2}{3 l n ( 2 )}}}{9 ln ( 2 )} d u

Take the constant out:

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

= \frac{1}{9 ln ( 2 )} \cdot \int u^{\frac{2}{3 l n ( 2 )}} d u

Apply the Power Rule

: \frac{3}{2 + 3 ln ( 2 )} ln (2) u^{\frac{2 + 3 l n ( 2 )}{3 l n ( 2 )}}

= \frac{1}{9 ln ( 2 )} \cdot \frac{3}{2 + 3 ln ( 2 )} ln (2) u^{\frac{2 + 3 l n ( 2 )}{3 l n ( 2 )}}

Substitute back

u = 51 2^{x}

= \frac{1}{9 ln ( 2 )} \cdot \frac{3}{2 + 3 ln ( 2 )} ln (2) (51 2^{x})^{\frac{2 + 3 l n ( 2 )}{3 l n ( 2 )}}

Simplify

\frac{1}{9 ln ( 2 )} \cdot \frac{3}{2 + 3 ln ( 2 )} ln (2) (51 2^{x})^{\frac{2 + 3 l n ( 2 )}{3 l n ( 2 )}} : \frac{1}{3 ( 3 ln ( 2 ) + 2 )} \cdot 51 2^{\frac{( 2 + 3 l n ( 2 ) ) x}{3 l n ( 2 )}}

= \frac{1}{3 ( 3 ln ( 2 ) + 2 )} \cdot 51 2^{\frac{( 2 + 3 l n ( 2 ) ) x}{3 l n ( 2 )}}

Add a constant to the solution

= \frac{1}{3 ( 3 ln ( 2 ) + 2 )} \cdot 51 2^{\frac{( 2 + 3 l n ( 2 ) ) x}{3 l n ( 2 )}} + C

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What is the integral of e^{6x}2^{9x} ?
The integral of e^{6x}2^{9x} is 1/(3(3ln(2)+2))*512^{((2+3ln(2))x)/(3ln(2))}+C