Q. 46 - Calculus

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.  

$\int 3x^2\sin \left(x^3+1\right) dx$

Step-by-Step Solution

Answer

The solution of the integral is $-\cos \left(x^3+1\right)+C.$

Step 1. Given Information.

The given integral is $\int 3x^2\sin \left(x^3+1\right) dx.$

Step 2. Solve.

By solving the integral we get, 

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

Step 3. Verification.

To verify the answer we differentiate $-\cos \left(x^3+1\right)+C$ it.

On differentiating we get,

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

Hence proved.