What is the integral of (3x^2)/(b^{5x^3)} ?

The integral of (3x^2)/(b^{5x^3)} is -1/(5b^{5x^3)ln(b)}+C

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integral of (3x^2)/(b^{5x^3)}

\int \frac{3 x ^{2}}{b ^{5 x^{3}}} d x

- \frac{1}{5 b ^{5 x^{3}} ln ( b )} + C

Solution steps

\int \frac{3 x ^{2}}{b ^{5 x^{3}}} d x

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{15 u ^{2} ln ( b )} d u

= 3 \cdot \int \frac{1}{15 u ^{2} ln ( b )} d u

Take the constant out:

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

= 3 \cdot \frac{1}{15 ln ( b )} \cdot \int \frac{1}{u ^{2}} d u

Apply the Power Rule

: - \frac{1}{u}

= 3 \cdot \frac{1}{15 ln ( b )} (- \frac{1}{u})

Substitute back

u = b^{5 x^{3}}

= 3 \cdot \frac{1}{15 ln ( b )} (- \frac{1}{b ^{5 x^{3}}})

Simplify

3 \cdot \frac{1}{15 ln ( b )} (- \frac{1}{b ^{5 x^{3}}}) : - \frac{1}{5 b ^{5 x^{3}} ln ( b )}

= - \frac{1}{5 b ^{5 x^{3}} ln ( b )}

Add a constant to the solution

= - \frac{1}{5 b ^{5 x^{3}} ln ( b )} + C

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What is the integral of (3x^2)/(b^{5x^3)} ?
The integral of (3x^2)/(b^{5x^3)} is -1/(5b^{5x^3)ln(b)}+C