Q. 35 - 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.  

$\int 3{\tan }^{2}x dx$

Step-by-Step Solution

Answer

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

Step 1. Given Information.

The given integral is $\int 3{\tan }^{2}x dx.$

Step 2. Solve.

By solving the integral we get, 

$$\begin{gathered} \int 3{\tan }^{2}x dx \\ =3\int {\tan }^{2}x dx \\ =3\int se{c}^{2}x-1 dx \\ =3\int se{c}^{2}x-3\int dx \\ =3 \tan \left(x\right)-3x+C \\ =3\left(\tan x-x\right)+C \end{gathered}$$

Step 3. Verification.

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

On differentiating we get,

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

Hence proved.