What is the integral of 1/2 t^2e^{t^3} ?

The integral of 1/2 t^2e^{t^3} is 1/6 e^{t^3}+C

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integral of 1/2 t^2e^{t^3}

\int \frac{1}{2} t^{2} e^{t^{3}} d t

\frac{1}{6} e^{t^{3}} + C

Solution steps

\int \frac{1}{2} t^{2} e^{t^{3}} d t

Take the constant out:

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

= \frac{1}{2} \cdot \int t^{2} e^{t^{3}} d t

Apply u-substitution

: \int \frac{e ^{u^{\frac{3}{2}}} u}{2 t} d u

= \frac{1}{2} \cdot \int \frac{e ^{u}}{3} d u

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{3} \cdot \int e^{u} d u

Use the common integral:

\int e^{u} d u = e^{u}

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

Substitute back

u = t^{3}

= \frac{1}{2} \cdot \frac{1}{3} e^{t^{3}}

Simplify

\frac{1}{2} \cdot \frac{1}{3} e^{t^{3}} : \frac{1}{6} e^{t^{3}}

= \frac{1}{6} e^{t^{3}}

Add a constant to the solution

= \frac{1}{6} e^{t^{3}} + C

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What is the integral of 1/2 t^2e^{t^3} ?
The integral of 1/2 t^2e^{t^3} is 1/6 e^{t^3}+C