What is the integral of (3x+1)^7 ?

The integral of (3x+1)^7 is 1/24 (3x+1)^8+C

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integral of (3x+1)^7

\int (3 x + 1)^{7} d x

\frac{1}{24} (3 x + 1)^{8} + C

Solution steps

\int (3 x + 1)^{7} d x

Apply u-substitution

= \int \frac{u ^{7}}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int u^{7} d u

Apply the Power Rule

= \frac{1}{3} \cdot \frac{u ^{8}}{8}

Substitute back

u = 3 x + 1

= \frac{1}{3} \cdot \frac{( 3 x + 1 ) ^{8}}{8}

Simplify

\frac{1}{3} \cdot \frac{( 3 x + 1 ) ^{8}}{8} : \frac{1}{24} (3 x + 1)^{8}

= \frac{1}{24} (3 x + 1)^{8}

Add a constant to the solution

= \frac{1}{24} (3 x + 1)^{8} + C