What is the integral of sqrt(1+4x^3)(12x^2) ?

The integral of sqrt(1+4x^3)(12x^2) is 2/3 (4x^3+1)^{3/2}+C

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integral of sqrt(1+4x^3)(12x^2)

\int 1 + 4 x^{3} (12 x^{2}) d x

\frac{2}{3} (4 x^{3} + 1)^{\frac{3}{2}} + C

Solution steps

\int 1 + 4 x^{3} \cdot 12 x^{2} d x

Take the constant out:

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

= 12 \cdot \int 1 + 4 x^{3} x^{2} d x

Apply u-substitution

= 12 \cdot \int \frac{u ^{2}}{6} d u

Take the constant out:

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

= 12 \cdot \frac{1}{6} \cdot \int u^{2} d u

Apply the Power Rule

= 12 \cdot \frac{1}{6} \cdot \frac{u ^{3}}{3}

Substitute back

u = 1 + 4 x^{3}

= 12 \cdot \frac{1}{6} \cdot \frac{( 1 + 4 x ^{3} ) ^{3}}{3}

Simplify

12 \cdot \frac{1}{6} \cdot \frac{( 1 + 4 x ^{3} ) ^{3}}{3} : \frac{2}{3} (4 x^{3} + 1)^{\frac{3}{2}}

= \frac{2}{3} (4 x^{3} + 1)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{3} (4 x^{3} + 1)^{\frac{3}{2}} + C

What is the integral of sqrt(1+4x^3)(12x^2) ?
The integral of sqrt(1+4x^3)(12x^2) is 2/3 (4x^3+1)^{3/2}+C