Q. 34 - 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.  

$\int 3csc\left(\pi x\right) cot\left(\pi x\right) dx$

Step-by-Step Solution

Answer

The solution of the integral is $-\frac{3}{\mathrm{\pi }}\cos \left(\mathrm{\pi x}\right)+C.$

Step 1. Given Information.

The given integral is$\int 3csc\left(\pi x\right) cot\left(\pi x\right) dx.$

Step 2. Solve.

By solving the integral we get, 

$$\begin{gathered} \int 3csc\left(\pi x\right) cot\left(\pi x\right) dx \\ =3\int csc\left(\pi x\right) cot\left(\pi x\right) dx \\ =3\int csc\left(\pi x\right) \frac{\cos \left(\pi x\right)}{\sin \left(\mathrm{\pi x}\right)} dx \\ =3\int cs{c}^2\left(\pi x\right) \cos \left(\pi x\right) dx \\ =3\left(\frac{1}{\mathrm{\pi }}\right) \left(\left(-cot\left(\mathrm{\pi x}\right)\right) \sin \left(\mathrm{\pi x}\right)\right)+C \\ =\frac{3}{\mathrm{\pi }}\left(\left(-\frac{\cos \left(\mathrm{\pi x}\right)}{\sin \left(\mathrm{\pi x}\right)}\right)\sin \left(\mathrm{\pi x}\right)\right)+C \\ =-\frac{3}{\mathrm{\pi }}\cos \left(\mathrm{\pi x}\right)+C \end{gathered}$$

Step 3. Verification.

To verify the answer we differentiate $-\frac{3}{\mathrm{\pi }}\cos \left(\mathrm{\pi x}\right)+C$ it.

On differentiating we get,

$$\begin{gathered} -\frac{3}{\mathrm{\pi }}\cos \left(\mathrm{\pi x}\right)+C \\ =-\frac{d}{dx}\left(\frac{3}{\mathrm{\pi }}\cos \left(\mathrm{\pi x}\right)\right)+\frac{d}{dx}\left(C\right) \\ =-\left(-3 csc\left(\mathrm{\pi x}\right) cot\left(\mathrm{\pi x}\right)\right) +0 \\ =3 csc\left(\mathrm{\pi x}\right) cot\left(\mathrm{\pi x}\right) \end{gathered}$$

Hence proved.