What is the integral of (7x+3)^5 ?

The integral of (7x+3)^5 is 1/42 (7x+3)^6+C

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integral of (7x+3)^5

\int (7 x + 3)^{5} d x

\frac{1}{42} (7 x + 3)^{6} + C

Solution steps

\int (7 x + 3)^{5} d x

Apply u-substitution

: \int \frac{u ^{5}}{7} d u

= \int \frac{u ^{5}}{7} d u

Take the constant out:

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

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

Apply the Power Rule

: \frac{u ^{6}}{6}

= \frac{1}{7} \cdot \frac{u ^{6}}{6}

Substitute back

u = 7 x + 3

= \frac{1}{7} \cdot \frac{( 7 x + 3 ) ^{6}}{6}

Simplify

\frac{1}{7} \cdot \frac{( 7 x + 3 ) ^{6}}{6} : \frac{1}{42} (7 x + 3)^{6}

= \frac{1}{42} (7 x + 3)^{6}

Add a constant to the solution

= \frac{1}{42} (7 x + 3)^{6} + C

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