What is the integral of x^23^{5x^3+1} ?

The integral of x^23^{5x^3+1} is (243^{x^3})/(5ln(3))+C

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integral of x^23^{5x^3+1}

\int x^{2} 3^{5 x^{3} + 1} d x

\frac{24 3 ^{x^{3}}}{5 ln ( 3 )} + C

Solution steps

\int x^{2} \cdot 3^{5 x^{3} + 1} d x

Apply u-substitution

: \int \frac{3 ^{5 u^{\frac{3}{2}} + 1} u}{2 x} d u

= \int 24 3^{u} d u

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

= \frac{24 3 ^{u}}{ln ( 243 )}

Substitute back

u = x^{3}

= \frac{24 3 ^{x^{3}}}{ln ( 243 )}

Simplify

ln (243) : 5 ln (3)

= \frac{24 3 ^{x^{3}}}{5 ln ( 3 )}

Add a constant to the solution

= \frac{24 3 ^{x^{3}}}{5 ln ( 3 )} + C

\int_{0}^{\frac{1}{2}} x^{2} d x

\int 2 y d y

\frac{\partial}{\partial x} ((x - 5)^{\frac{1}{4}} y^{\frac{1}{4}})

f (x) = e^{x} sin (x)

x \to 1 lim (x^{2} + 3)

What is the integral of x^23^{5x^3+1} ?
The integral of x^23^{5x^3+1} is (243^{x^3})/(5ln(3))+C