Answer

Verified

338.1k+ views

**Hint:**Find the integration of the given function in order to find its antiderivative. The given trigonometric expression is non elementary, so use the Maclaurin power expansion of the given function and then integrate that expansion to the antiderivative of the given function. Maclaurin expansion of cosine is given as

$\cos x = \sum\limits_{n = 0}^\infty {\dfrac{{{{( - 1)}^n}}}{{(2n)!}}{x^{2n}}} $

Use this information to find the antiderivative of the given function.

**Formula used:**

Maclaurin series of cosine $\cos x = \sum\limits_{n = 0}^\infty {\dfrac{{{{( - 1)}^n}}}{{(2n)!}}{x^{2n}}} $

**Complete step by step solution:**

To find the antiderivative of $\cos \left( {{x^2}} \right)$, let us understand first what is antiderivative of a function?

When we take an antiderivative also called the inverse derivative of a function f is a function then it gives a function F whose derivative is equal to the original function f.

That is in simple words antiderivative is the integration of a function.

Therefore to find the antiderivative of $\cos \left( {{x^2}} \right)$, we will find its integration.

Since the integral of $\cos \left( {{x^2}} \right)$ is non elementary, we cannot integrate it directly, so we will integrate it via power series.

For integration via power series, let us recall the Maclaurin series of cosine,

$

\cos x = \sum\limits_{n = 0}^\infty {\dfrac{{{{( - 1)}^n}}}{{(2n)!}}{x^{2n}}} \\

\Rightarrow \cos \left( {{x^2}} \right) = \sum\limits_{n = 0}^\infty {\dfrac{{{{( -

1)}^n}}}{{(2n)!}}{{\left( {{x^2}} \right)}^{2n}}} \\

\therefore \cos \left( {{x^2}} \right) = \sum\limits_{n = 0}^\infty {\dfrac{{{{( -

1)}^n}}}{{(2n)!}}{x^{4n}}} \\

$

Taking the integration both sides, we will get

$\int {\cos \left( {{x^2}} \right)dx} = \int {\sum\limits_{n = 0}^\infty {\dfrac{{{{( -

1)}^n}}}{{(2n)!}}{x^{4n}}} } dx$

Here taking out the constant term,

$

\int {\cos \left( {{x^2}} \right)dx} = \sum\limits_{n = 0}^\infty {\dfrac{{{{( - 1)}^n}}}{{(2n)!}}} \int

{{x^{4n}}} dx \\

= \sum\limits_{n = 0}^\infty {\dfrac{{{{( - 1)}^n}}}{{(2n)!}}} \times \dfrac{{{x^{4n + 1}}}}{{(4n + 1)}} +

C \\

= \sum\limits_{n = 0}^\infty {\dfrac{{{{( - 1)}^n}{x^{4n + 1}}}}{{(2n)!(4n + 1)}}} + C \\

$

**Therefore $\sum\limits_{n = 0}^\infty {\dfrac{{{{( - 1)}^n}{x^{4n + 1}}}}{{(2n)!(4n + 1)}}} + C$ is the required antiderivative of $\cos \left( {{x^2}} \right)$**

**Note:**There is a bit difference in antiderivative and integration, Integration is a function associates with the original function whereas antiderivative of $f(x)$ is just a function whose derivative is $f(x)$ This question can be solved with one more method in which we integrate $\cos \left( {{x^2}} \right)$ with help of the Fresnel integral. You will get a different expression at the end of the Fresnel integral process for this question but don’t worry both are equal and correct.

Recently Updated Pages

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE

Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE

What are the possible quantum number for the last outermost class 11 chemistry CBSE

Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE

What happens when entropy reaches maximum class 11 chemistry JEE_Main

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Write a stanza wise summary of money madness class 11 english CBSE

Which places in India experience sunrise first and class 9 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Difference Between Plant Cell and Animal Cell

Which neighbouring country does not share a boundary class 9 social science CBSE

What is Whales collective noun class 10 english CBSE