What is the integral of 5x^2b^{7x^3} ?

The integral of 5x^2b^{7x^3} is (5b^{7x^3})/(21ln(b))+C

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integral of 5x^2b^{7x^3}

\int 5 x^{2} b^{7 x^{3}} d x

\frac{5 b ^{7 x^{3}}}{21 ln ( b )} + C

Solution steps

\int 5 x^{2} b^{7 x^{3}} d x

Take the constant out:

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

= 5 \cdot \int x^{2} b^{7 x^{3}} d x

Apply u-substitution

: \int \frac{b ^{7 u^{\frac{3}{2}}} u}{2 x} d u

= 5 \cdot \int \frac{b ^{7 u}}{3} d u

Take the constant out:

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

= 5 \cdot \frac{1}{3} \cdot \int b^{7 u} d u

Apply u-substitution

: \int b^{v} \frac{1}{7} d v

= 5 \cdot \frac{1}{3} \cdot \int b^{v} \frac{1}{7} d v

Take the constant out:

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

= 5 \cdot \frac{1}{3} \cdot \frac{1}{7} \cdot \int b^{v} d v

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

= 5 \cdot \frac{1}{3} \cdot \frac{1}{7} \cdot \frac{b ^{v}}{ln ( b )}

Substitute back

= 5 \cdot \frac{1}{3} \cdot \frac{1}{7} \cdot \frac{b ^{7 x^{3}}}{ln ( b )}

Simplify

5 \cdot \frac{1}{3} \cdot \frac{1}{7} \cdot \frac{b ^{7 x^{3}}}{ln ( b )} : \frac{5 b ^{7 x^{3}}}{21 ln ( b )}

= \frac{5 b ^{7 x^{3}}}{21 ln ( b )}

Add a constant to the solution

= \frac{5 b ^{7 x^{3}}}{21 ln ( b )} + C

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What is the integral of 5x^2b^{7x^3} ?
The integral of 5x^2b^{7x^3} is (5b^{7x^3})/(21ln(b))+C