Answer

Verified

406.8k+ views

**Hint:**

Here, we will first rationalize the given numerator and try to simplify it further so that we can apply the u-substitution method. Solving this further and using the formula of integration of ${x^n}$, we will be able to find the required answer. Integration is defined as the summation of all the discrete data.

**Formula Used:**

We will use the following formulas:

1) $\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}$

2) ${\sin ^2}x + {\cos ^2}x = 1$

3) $\int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}}} + C$

**Complete step by step solution:**

The given integral function is $\int {\left( {\dfrac{{\sqrt {1 - \cos x} }}{2}} \right)} dx$.

This can also be written as: $\dfrac{1}{2}\int {\sqrt {1 - \cos x} } dx$

Now, we will do the rationalization by multiplying and dividing by $\sqrt {1 + \cos x} $

Hence, we get,

$\dfrac{1}{2}\int {\sqrt {1 - \cos x} } dx = \dfrac{1}{2}\int {\dfrac{{\sqrt {1 - \cos x} \times \sqrt {1 + \cos x} }}{{\sqrt {1 + \cos x} }}} dx$

Using the identity $\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}$ in the numerator, we get,

$ \Rightarrow \dfrac{1}{2}\int {\sqrt {1 - \cos x} } dx = \dfrac{1}{2}\int {\dfrac{{\sqrt {1 - {{\cos }^2}x} }}{{\sqrt {1 + \cos x} }}} dx$

Also, ${\sin ^2}x + {\cos ^2}x = 1$

Or $1 - {\cos ^2}x = {\sin ^2}x$

Thus, substituting this value in the numerator,

$ \Rightarrow \dfrac{1}{2}\int {\sqrt {1 - \cos x} } dx = \dfrac{1}{2}\int {\dfrac{{\sqrt {{{\sin }^2}x} }}{{\sqrt {1 + \cos x} }}} dx = \dfrac{1}{2}\int {\dfrac{{\sin x}}{{\sqrt {1 + \cos x} }}} dx$…………………..$\left( 1 \right)$

Now, by u-substitution, let $1 + \cos x = t$

Differentiating both sides, with respect to $x$, we get,

$ - \sin xdx = dt$

Or $\sin xdx = - dt$

Substituting this in $\left( 1 \right)$, we get,

$ \Rightarrow \dfrac{1}{2}\int {\sqrt {1 - \cos x} } dx = - \dfrac{1}{2}\int {\dfrac{1}{{\sqrt t }}} dt$

This can also be written as:

$ \Rightarrow \dfrac{1}{2}\int {\sqrt {1 - \cos x} } dx = - \dfrac{1}{2}\int {{t^{ - \dfrac{1}{2}}}} dt$

We know that, $\int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}}} + C$

Hence, we get,

$ \Rightarrow \dfrac{1}{2}\int {\sqrt {1 - \cos x} } dx = - \dfrac{1}{2}\left( {\dfrac{{{t^{ - \dfrac{1}{2} + 1}}}}{{ - \dfrac{1}{2} + 1}}} \right) + C = - \dfrac{1}{2}\left( {\dfrac{{{t^{\dfrac{1}{2}}}}}{{\dfrac{1}{2}}}} \right) + C$

$ \Rightarrow \dfrac{1}{2}\int {\sqrt {1 - \cos x} } dx = - 2 \times \dfrac{1}{2}\left( {{t^{\dfrac{1}{2}}}} \right) + C = - \sqrt t + C$

But we know that, $1 + \cos x = t$

Hence,

**$\int {\left( {\dfrac{{\sqrt {1 - \cos x} }}{2}} \right)} dx = - \sqrt {1 + \cos x} + C$**

Therefore, this is the required answer.

Therefore, this is the required answer.

**Note:**

In calculus, integration by substitution, also known as u-substitution or change of variables, is a method which is used for evaluating integrals or anti-derivatives. It is the counterpart to the chain rule for differentiation, in fact, it can also be considered as doing the chain rule "backwards". Also, there is another method which is known as integration by parts which is a process of finding the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. But, clearly, in this question, we didn’t have any product of functions thus, we cannot use the by parts method.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

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

Trending doubts

Which are the Top 10 Largest Countries of the World?

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

How do you graph the function fx 4x class 9 maths CBSE

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

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE