What is the integral of (3x+1)^{11} ?

The integral of (3x+1)^{11} is 1/36 (3x+1)^{12}+C

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integral of (3x+1)^{11}

\int (3 x + 1)^{11} d x

\frac{1}{36} (3 x + 1)^{12} + C

Solution steps

\int (3 x + 1)^{11} d x

Apply u-substitution

: \int \frac{u ^{11}}{3} d u

= \int \frac{u ^{11}}{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^{11} d u

Apply the Power Rule

: \frac{u ^{12}}{12}

= \frac{1}{3} \cdot \frac{u ^{12}}{12}

Substitute back

u = 3 x + 1

= \frac{1}{3} \cdot \frac{( 3 x + 1 ) ^{12}}{12}

Simplify

\frac{1}{3} \cdot \frac{( 3 x + 1 ) ^{12}}{12} : \frac{1}{36} (3 x + 1)^{12}

= \frac{1}{36} (3 x + 1)^{12}

Add a constant to the solution

= \frac{1}{36} (3 x + 1)^{12} + C

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What is the integral of (3x+1)^{11} ?
The integral of (3x+1)^{11} is 1/36 (3x+1)^{12}+C