What is the integral of 3r^2e^{r^3} ?

The integral of 3r^2e^{r^3} is e^{r^3}+C

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integral of 3r^2e^{r^3}

\int 3 r^{2} e^{r^{3}} d r

e^{r^{3}} + C

Solution steps

\int 3 r^{2} e^{r^{3}} d r

Take the constant out:

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

= 3 \cdot \int r^{2} e^{r^{3}} d r

Apply u-substitution

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

= 3 \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

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

Use the common integral:

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

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

Substitute back

u = r^{3}

= 3 \cdot \frac{1}{3} e^{r^{3}}

Simplify

3 \cdot \frac{1}{3} e^{r^{3}} : e^{r^{3}}

= e^{r^{3}}

Add a constant to the solution

= e^{r^{3}} + C

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