What is the integral of (12(3x+1)^3) ?

The integral of (12(3x+1)^3) is (3x+1)^4+C

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integral of (12(3x+1)^3)

\int (12 (3 x + 1)^{3}) d x

(3 x + 1)^{4} + C

Solution steps

\int 12 (3 x + 1)^{3} d x

Take the constant out:

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

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

Apply u-substitution

= 12 \cdot \int \frac{u ^{3}}{3} d u

Take the constant out:

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

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

Apply the Power Rule

= 12 \cdot \frac{1}{3} \cdot \frac{u ^{4}}{4}

Substitute back

u = 3 x + 1

= 12 \cdot \frac{1}{3} \cdot \frac{( 3 x + 1 ) ^{4}}{4}

Simplify

12 \cdot \frac{1}{3} \cdot \frac{( 3 x + 1 ) ^{4}}{4} : (3 x + 1)^{4}

= (3 x + 1)^{4}

Add a constant to the solution

= (3 x + 1)^{4} + C

What is the integral of (12(3x+1)^3) ?
The integral of (12(3x+1)^3) is (3x+1)^4+C