What is the integral of t^3sqrt(2t^4+3) ?

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

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integral of t^3sqrt(2t^4+3)

\int t^{3} 2 t^{4} + 3 d t

\frac{1}{12} (2 t^{4} + 3)^{\frac{3}{2}} + C

Solution steps

\int t^{3} 2 t^{4} + 3 d t

Apply u-substitution

= \int \frac{u}{8} d u

Take the constant out:

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

= \frac{1}{8} \cdot \int u d u

Apply the Power Rule

= \frac{1}{8} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 2 t^{4} + 3

= \frac{1}{8} \cdot \frac{2}{3} (2 t^{4} + 3)^{\frac{3}{2}}

Simplify

\frac{1}{8} \cdot \frac{2}{3} (2 t^{4} + 3)^{\frac{3}{2}} : \frac{1}{12} (2 t^{4} + 3)^{\frac{3}{2}}

= \frac{1}{12} (2 t^{4} + 3)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{12} (2 t^{4} + 3)^{\frac{3}{2}} + C

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