What is the integral of 8x^3(2x^4+56)^6 ?

The integral of 8x^3(2x^4+56)^6 is 1/7 (2x^4+56)^7+C

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integral of 8x^3(2x^4+56)^6

\int 8 x^{3} (2 x^{4} + 56)^{6} d x

\frac{1}{7} (2 x^{4} + 56)^{7} + C

Solution steps

\int 8 x^{3} (2 x^{4} + 56)^{6} d x

Take the constant out:

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

= 8 \cdot \int x^{3} (2 x^{4} + 56)^{6} d x

Apply u-substitution

: \int \frac{u ^{6}}{8} d u

= 8 \cdot \int \frac{u ^{6}}{8} d u

Take the constant out:

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

= 8 \cdot \frac{1}{8} \cdot \int u^{6} d u

Apply the Power Rule

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

= 8 \cdot \frac{1}{8} \cdot \frac{u ^{7}}{7}

Substitute back

u = 2 x^{4} + 56

= 8 \cdot \frac{1}{8} \cdot \frac{( 2 x ^{4} + 56 ) ^{7}}{7}

Simplify

8 \cdot \frac{1}{8} \cdot \frac{( 2 x ^{4} + 56 ) ^{7}}{7} : \frac{1}{7} (2 x^{4} + 56)^{7}

= \frac{1}{7} (2 x^{4} + 56)^{7}

Add a constant to the solution

= \frac{1}{7} (2 x^{4} + 56)^{7} + C

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What is the integral of 8x^3(2x^4+56)^6 ?
The integral of 8x^3(2x^4+56)^6 is 1/7 (2x^4+56)^7+C