What is the integral of 3x(5x^2+3)^6 ?

The integral of 3x(5x^2+3)^6 is 3/70 (5x^2+3)^7+C

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integral of 3x(5x^2+3)^6

\int 3 x (5 x^{2} + 3)^{6} d x

\frac{3}{70} (5 x^{2} + 3)^{7} + C

Solution steps

\int 3 x (5 x^{2} + 3)^{6} d x

Take the constant out:

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

= 3 \cdot \int x (5 x^{2} + 3)^{6} d x

Apply u-substitution

: \int \frac{u ^{6}}{10} d u

= 3 \cdot \int \frac{u ^{6}}{10} d u

Take the constant out:

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

= 3 \cdot \frac{1}{10} \cdot \int u^{6} d u

Apply the Power Rule

: \frac{u ^{7}}{7}

= 3 \cdot \frac{1}{10} \cdot \frac{u ^{7}}{7}

Substitute back

u = 5 x^{2} + 3

= 3 \cdot \frac{1}{10} \cdot \frac{( 5 x ^{2} + 3 ) ^{7}}{7}

Simplify

3 \cdot \frac{1}{10} \cdot \frac{( 5 x ^{2} + 3 ) ^{7}}{7} : \frac{3}{70} (5 x^{2} + 3)^{7}

= \frac{3}{70} (5 x^{2} + 3)^{7}

Add a constant to the solution

= \frac{3}{70} (5 x^{2} + 3)^{7} + C

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What is the integral of 3x(5x^2+3)^6 ?
The integral of 3x(5x^2+3)^6 is 3/70 (5x^2+3)^7+C