Question

Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess-and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating.

$\int x^3{e}^{3x^4-2}dx$

Short answer

The solution of the integral is$\frac{1}{12}{e}^{3x^4-2}+C.$

Step-by-step solution

Step 1. Given Information.  

The given integral is$\int x^3{e}^{3x^4-2}dx.$

Step 2. Solve. 

By solving the integral we get,

$$\begin{gathered} \int x^3{e}^{3x^4-2}dx \\ \mathrm{Let}u=3x^4-2,du=12x^3 \\ =\int \left(\frac{1}{12}\right)e^{u}du \\ =\frac{1}{12}\int e^{u}du \\ =\frac{1}{12}{e}^u+C \\ \mathrm{Substitute}\mathrm{back}u=3x^4-2 \\ =\frac{1}{12}{e}^{3x^4-2}+C \end{gathered}$$

Step 3. Verification. 

To verify the answer we differentiate $\frac{1}{12}{e}^{3x^4-2}+C$it.

On differentiating we get,

$$\begin{gathered} \frac{1}{12}{e}^{3x^4-2}+C \\ =\frac{d}{dx}\frac{1}{12}{e}^{3x^4-2}+\frac{d}{dx}C \\ =\frac{1}{12}\frac{d}{dx}{e}^{3x^4-2}+\frac{d}{dx}C \\ =\frac{1}{12}\left(12x^3\right)e^{3x^4-2}+0 \\ =x^3{e}^{3x^4-2} \end{gathered}$$

Hence proved.