What is the integral of (7-5x)^6 ?

The integral of (7-5x)^6 is -1/35 (7-5x)^7+C

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integral of (7-5x)^6

\int (7 - 5 x)^{6} d x

- \frac{1}{35} (7 - 5 x)^{7} + C

Solution steps

\int (7 - 5 x)^{6} d x

Apply u-substitution

: \int - \frac{u ^{6}}{5} d u

= \int - \frac{u ^{6}}{5} d u

Take the constant out:

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

= - \frac{1}{5} \cdot \int u^{6} d u

Apply the Power Rule

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

= - \frac{1}{5} \cdot \frac{u ^{7}}{7}

Substitute back

u = 7 - 5 x

= - \frac{1}{5} \cdot \frac{( 7 - 5 x ) ^{7}}{7}

Simplify

- \frac{1}{5} \cdot \frac{( 7 - 5 x ) ^{7}}{7} : - \frac{1}{35} (7 - 5 x)^{7}

= - \frac{1}{35} (7 - 5 x)^{7}

Add a constant to the solution

= - \frac{1}{35} (7 - 5 x)^{7} + C

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