Q. 20 - Calculus

Question

Indefinite integrals: Use integration formulas, algebra, and educated guess-and-check strategies to find the following integrals.

$\int 2\cos \left(3x\right)dx$

Step-by-Step Solution

Answer

The solution is: $\frac{2}{3}\sin \left(3x\right)+C$

Step 1: Given

The given integral is: $\int 2\cos \left(3x\right)dx$

Step 2: To Find

Use integration formulas, algebra, and educated guess-and-check strategies to find the given integral.

Step 3: Calculation

Let 

$$\begin{gathered} u=3x \\ du=3dx \\ \frac{du}{3}=dx \end{gathered}$$

Using substitution we get:

$$\begin{gathered} \int 2\cos \left(3x\right)dx \\ =\int 2\cos \left(u\right)\frac{du}{3} \\ =\frac{2}{3}\int \cos \left(u\right)du \\ =\frac{2}{3}\sin \left(u\right)+C \\ =\frac{2}{3}\sin \left(3x\right)+C \end{gathered}$$

Where, $C$= integrating constant.