Question

How do I find the indefinite integral of $\sin (3x)\cos (3x)?$

Answer 1

In this question, you have to find the indefinite integral of the given trigonometric function. To find the integral, firstly use the trigonometric identity for half angle formula of sine function to simplify the given trigonometric function then integrate the simplified function with help of basic integration formulas for trigonometry.Half angle formula for sine function is given as, $\sin 2x = 2\sin x\cos x$.

Complete step by step answer:
In order to find the indefinite integral of the given trigonometric function $\sin (3x)\cos (3x)$ we will first simplify the given trigonometric function with help of half angle formula for sine function as follows,
$\sin (3x)\cos (3x)$
Dividing and multiplying the above expression with two, we will get
$\dfrac{1}{2} \times 2\sin (3x)\cos (3x)$
Now, we can see that the above expression is looking similar to the trigonometric identity of half angle formula for sine function that is given as $\sin 2x = 2\sin x\cos x$, so using this we can further simplify the expression as
$\dfrac{1}{2} \times \sin (6x)$
Now integrating it, we will get
$\int {\dfrac{1}{2} \times \sin (6x)dx} \\\ \Rightarrow \dfrac{1}{2} \times \int {\sin (6x)dx} \\\ \Rightarrow \dfrac{1}{2} \times \left( {\dfrac{{ - \cos (6x)}}{6} + c} \right) \\\$
Simplifying this, we will get
$\dfrac{{ - \cos (6x)}}{{12}} + \dfrac{1}{2} \times c \\\ \therefore\dfrac{{ - \cos (6x)}}{{12}} + C \\\$
Therefore $\dfrac{{ - \cos (6x)}}{{12}} + C$ is the required indefinite integral of the given function.

Additional information:
We have written $\dfrac{1}{2} \times c = C$ in the integral part because the constant we get when doing the indefinite integration is an arbitrary constant and we can consider anything to be an arbitrary constant and also algebraic operation does not affect arbitrary constants.

Note: When integrating a term consisting of a constant then you can write the constant outside the integral as we have written in this question. Also take care of writing the constant part after doing indefinite integral.