What is the integral of (-x^8+4)^6x^7 ?

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

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integral of (-x^8+4)^6x^7

\int (- x^{8} + 4)^{6} x^{7} d x

- \frac{1}{56} (- x^{8} + 4)^{7} + C

Solution steps

\int (- x^{8} + 4)^{6} x^{7} d x

Apply u-substitution

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

= \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

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

Apply the Power Rule

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

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

Substitute back

u = - x^{8} + 4

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

Simplify

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

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

Add a constant to the solution

= - \frac{1}{56} (- x^{8} + 4)^{7} + C

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