Question

How do you integrate this ${\int {\left( {\cos 2x} \right)} ^2}dx$ ?

Answer 1

Hint: To solve and take out the integral of this question , we need to solve it step by step . Here we are going to perform some calculations and formulae of integration to simplify the given question and then again adding back together when the individual fraction would be solved to make our question solve easily . Some basic trigonometry formulae will come into existence while solving the integration part and Don’t forget to place Constant of integration $C$at the end of the integral.

Complete step-by-step solution:
We are given an expression ${\int {\left( {\cos 2x} \right)} ^2}dx$ and we have to calculate its integral .
Now we can apply here some basic trigonometry formulae which we know that of
 $\cos 2A = 2{\cos ^2}A - 1$
Or we can rewrite it as –
${\cos ^2}A = \dfrac{1}{2}(\cos 2A + 1)$
Now we are going to separate the constant fraction integrand to solve separately as if an integrand can be separated then all its parts can be solved separately . After separating we get –
\[
  \int {{{\cos }^2}(2x)} dx = \int {\dfrac{{1 + \cos (4x)}}{2}} \\
   \Rightarrow \dfrac{1}{2}\int {1 + } \cos (4x)dx \\
   \Rightarrow \dfrac{1}{2}\int {1dx + \int {\cos (4x)dx} } \\
   \Rightarrow \dfrac{1}{2}x + \int {\cos (4x)dx} \\
   \Rightarrow \dfrac{1}{2}\left[ {x + \left( {\dfrac{1}{4}\sin 4x} \right)} \right] \\
   \Rightarrow \dfrac{x}{2} + \dfrac{1}{8}\sin 4x + C \\
 \]
This is the required answer \[\dfrac{x}{2} + \dfrac{1}{8}\sin 4x + C\].
Additional Information:
1.Different types of methods of Integration:
>Integration by Substitution
>Integration by parts
>Integration of rational algebraic function by using partial fraction
2. Integration by Substitution: The method of evaluating the integral by reducing it to standard form by a proper substitution is called integration by substitution.
If $\varphi (x)$is a continuously differentiable function, then to evaluate integrals of the form.
\[\int {f(\varphi(x))\,{\varphi '}(x)dx} \], we substitute $\varphi (x)$=t and ${\varphi'}(x)dx = dt$
This substitution reduces the above integral to \[\int {f(t)\,dt} \]. After evaluating this integral we substitute back the value of t.

Note: Use standard formula carefully while evaluating the integrals.
Indefinite integral=Let $f(x)$ be a function .Then the family of all its primitives (or antiderivatives) is called the indefinite integral of $f(x)$ and is denoted by $\int {f(x)} dx$. The symbol $\int {f(x)dx} $ is read as the indefinite integral of $f(x)$ with respect to x. C is known as the constant of integration and Don’t forget to place Constant of integration $C$ at the end of the integral. Cross check the answer and always keep the final answer simplified .
Remember the trigonometric identities and apply appropriately .