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

The integral of (2(3+5x)^5)/5 is 1/75 (3+5x)^6+C

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

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

\frac{1}{75} (3 + 5 x)^{6} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

= \frac{2}{5} \cdot \int \frac{u ^{5}}{5} d u

Take the constant out:

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

= \frac{2}{5} \cdot \frac{1}{5} \cdot \int u^{5} d u

Apply the Power Rule

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

= \frac{2}{5} \cdot \frac{1}{5} \cdot \frac{u ^{6}}{6}

Substitute back

u = 3 + 5 x

= \frac{2}{5} \cdot \frac{1}{5} \cdot \frac{( 3 + 5 x ) ^{6}}{6}

Simplify

\frac{2}{5} \cdot \frac{1}{5} \cdot \frac{( 3 + 5 x ) ^{6}}{6} : \frac{1}{75} (3 + 5 x)^{6}

= \frac{1}{75} (3 + 5 x)^{6}

Add a constant to the solution

= \frac{1}{75} (3 + 5 x)^{6} + C

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What is the integral of (2(3+5x)^5)/5 ?
The integral of (2(3+5x)^5)/5 is 1/75 (3+5x)^6+C