What is the integral of 4x5^{3x^2} ?

The integral of 4x5^{3x^2} is (2*125^{x^2})/(3ln(5))+C

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integral of 4x5^{3x^2}

\int 4 x 5^{3 x^{2}} d x

\frac{2 \cdot 12 5 ^{x^{2}}}{3 ln ( 5 )} + C

Solution steps

\int 4 x \cdot 5^{3 x^{2}} d x

Take the constant out:

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

= 4 \cdot \int x \cdot 5^{3 x^{2}} d x

Apply u-substitution

: \int \frac{12 5 ^{u}}{2} d u

= 4 \cdot \int \frac{12 5 ^{u}}{2} d u

Take the constant out:

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

= 4 \cdot \frac{1}{2} \cdot \int 12 5^{u} d u

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

= 4 \cdot \frac{1}{2} \cdot \frac{12 5 ^{u}}{ln ( 125 )}

Substitute back

u = x^{2}

= 4 \cdot \frac{1}{2} \cdot \frac{12 5 ^{x^{2}}}{ln ( 125 )}

Simplify

4 \cdot \frac{1}{2} \cdot \frac{12 5 ^{x^{2}}}{ln ( 125 )} : \frac{2 \cdot 12 5 ^{x^{2}}}{3 ln ( 5 )}

= \frac{2 \cdot 12 5 ^{x^{2}}}{3 ln ( 5 )}

Add a constant to the solution

= \frac{2 \cdot 12 5 ^{x^{2}}}{3 ln ( 5 )} + C

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What is the integral of 4x5^{3x^2} ?
The integral of 4x5^{3x^2} is (2*125^{x^2})/(3ln(5))+C