Q. 28 - Calculus

Question

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.) 

$\int 3\cos \left(\pi x\right)\mathrm{d}x$

Step-by-Step Solution

Answer

The solution of the given integral is $\int 3\cos \left(\pi x\right)\mathrm{d}x=\frac{3}{\mathrm{\pi }}\mathrm{sin\pi x}+\mathrm{C}$.

Step 1. Given Information

Solving the given integrals. 

$\int 3\cos \left(\pi x\right)\mathrm{d}x$

Step 2. Solving the given integral using substitution method.

Let 

$$\begin{gathered} u=\pi x \\ \frac{du}{dx}=\pi \\ du=\pi dx \\ \frac{1}{\pi }du=dx \end{gathered}$$

Step 3. This substitution changes the integral into

$$\begin{gathered} \int 3\cos \left(\pi x\right)\mathrm{d}x=\frac{3}{\mathrm{\pi }}\int \cos u\mathrm{du} \\ \int 3\cos \left(\mathrm{\pi x}\right)\mathrm{dx}=\frac{3}{\mathrm{\pi }}\mathrm{sinu}+\mathrm{C} \\ \int 3\cos \left(\mathrm{\pi x}\right)\mathrm{dx}=\frac{3}{\mathrm{\pi }}\mathrm{sin\pi x}+\mathrm{C} \end{gathered}$$