What is the integral of (3x+1)^43 ?

The integral of (3x+1)^43 is 1/5 (3x+1)^5+C

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integral of (3x+1)^43

\int (3 x + 1)^{4} 3 d x

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

Solution steps

\int (3 x + 1)^{4} \cdot 3 d x

Take the constant out:

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

= 3 \cdot \int (3 x + 1)^{4} d x

Apply u-substitution

: \int \frac{u ^{4}}{3} d u

= 3 \cdot \int \frac{u ^{4}}{3} d u

Take the constant out:

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

= 3 \cdot \frac{1}{3} \cdot \int u^{4} d u

Apply the Power Rule

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

= 3 \cdot \frac{1}{3} \cdot \frac{u ^{5}}{5}

Substitute back

u = 3 x + 1

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

Simplify

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

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

Add a constant to the solution

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

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