What is the integral of 3t^2sqrt(t^3+2) ?

The integral of 3t^2sqrt(t^3+2) is 2/3 (t^3+2)^{3/2}+C

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integral of 3t^2sqrt(t^3+2)

\int 3 t^{2} t^{3} + 2 d t

\frac{2}{3} (t^{3} + 2)^{\frac{3}{2}} + C

Solution steps

\int 3 t^{2} t^{3} + 2 d t

Take the constant out:

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

= 3 \cdot \int t^{2} t^{3} + 2 d t

Apply u-substitution

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

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

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= 3 \cdot \frac{1}{3} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = t^{3} + 2

= 3 \cdot \frac{1}{3} \cdot \frac{2}{3} (t^{3} + 2)^{\frac{3}{2}}

Simplify

3 \cdot \frac{1}{3} \cdot \frac{2}{3} (t^{3} + 2)^{\frac{3}{2}} : \frac{2}{3} (t^{3} + 2)^{\frac{3}{2}}

= \frac{2}{3} (t^{3} + 2)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{3} (t^{3} + 2)^{\frac{3}{2}} + C

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What is the integral of 3t^2sqrt(t^3+2) ?
The integral of 3t^2sqrt(t^3+2) is 2/3 (t^3+2)^{3/2}+C