Q. 33 - 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.  

$\int sec\left(3x\right) \tan \left(3x\right) dx$

Step-by-Step Solution

Answer

The solution of the integral is $\frac{1}{3}sec 3x +C.$

Step 1. Given Information.

The given integral is $\int sec\left(3x\right) \tan \left(3x\right) dx.$

Step 2. Solve.

By solving the integral we get, 

$$\begin{gathered} \int sec\left(3x\right) \tan \left(3x\right) dx \\ \mathrm{Use} \int \sec \left(\mathrm{x}\right) \tan \left(\mathrm{x}\right) \mathrm{dx}=sec\left(x\right)+C \\ =\frac{1}{3}sec3x +C \end{gathered}$$

Step 3. Verification.

To verify the answer we differentiate $\frac{1}{3}sec3x +C$ it.

On differentiating we get,

$$\begin{gathered} \frac{1}{3}sec3x +C \\ =\frac{d}{dx}\left(\frac{1}{3}sec 3x\right)+\frac{d}{dx}C \\ =3\left(\frac{1}{3}sec\left(3x\right) \tan \left(3x\right)\right)+0 \\ =sec\left(3x\right) \tan \left(3x\right) \end{gathered}$$

Hence proved.