Question

Solve the integration $\int {\dfrac{1}{{\sqrt {7 - {x^2}} }}dx}$
$\left( A \right)$ $\dfrac{1}{{2\sqrt 7 }}\log \left| {\dfrac{{\sqrt {7 + x} }}{{\sqrt {7 - x} }}} \right| + c$
$\left( B \right)$ ${\sin ^{ - 1}}\dfrac{x}{{\sqrt 7 }} + c$
$\left( C \right)$ $\log \left| {x + \sqrt {{x^2} - 7} } \right| + c$
$\left( D \right)$ $\dfrac{1}{{2\sqrt 7 }}\log \left| {\dfrac{{x - \sqrt 7 }}{{x + \sqrt 7 }}} \right| + c$

Answer 1

We have to integrate the given integral expression with respect to $x$ . We solve this question using the concept of various formulas of integration , the concept of substitution method and we should also have the knowledge of the concept of various trigonometric identities . First we will substitute the value of $x$ with a function of sine such that the term of the interaction gets simplified , then using the formula of sum of square of sine and cosine function , we will further simplify the expression and thus on solving the integral and substituting back the value of the stetted term we will get the required value of the integral expression.

Complete step by step answer:
Given integral is $\int {\dfrac{1}{{\sqrt {7 - {x^2}} }}dx}$
Let us consider that $I = \int {\dfrac{1}{{\sqrt {7 - {x^2}} }}dx}$
Now , we have to integrate I with respect to $x$ .
Put $x = \sqrt 7 \sin a$
$a = {\sin ^{ - 1}}\dfrac{x}{{\sqrt 7 }}$
Now differentiating $x$ with respect to $a$ , we get
$dx = \sqrt 7 \cos a{\text{ }}da$
Substituting the values of $dx$ in the given integral , we can write the expression as :
$I = \int {\dfrac{{\sqrt 7 \cos a}}{{\sqrt {7 - {{\left( {\sqrt 7 \sin a} \right)}^2}} }}} da$
On further simplifying , we can write the expression as :
$I = \int {\dfrac{{\sqrt 7 \cos a}}{{\sqrt {7 - 7{{\sin }^2}a} }}} da$
Taking $7$ common from the denominator , we can write the expression as :
$I = \int {\dfrac{{\sqrt 7 \cos a}}{{\sqrt {7\left( {1 - {{\sin }^2}a} \right)} }}} da$
Now, we also know that the formula of sum of square of sine and cosine is given as :
${\sin ^2}a + {\cos ^2}a = 1$
${\cos ^2}a = 1 - {\sin ^2}a$
Putting the value , we can write the expression as :
$I = \int {\dfrac{{\sqrt 7 \cos a}}{{\sqrt {7{{\cos }^2}a} }}} da$
$I = \int {\dfrac{{\sqrt 7 \cos a}}{{\sqrt 7 \cos a}}} da$
On cancelling the terms , we can write the expression as :
$I = \int 1 da$
Also we know that the formula of integration of ${x^n}$ is given as :
$\int {{x^n}} = \dfrac{{{x^{n + 1}}}}{{n + 1}}$
Using the formula of integration , we can write the expression as :
$I = a + c$
Where $c$ is the integrating constant .
Now, substituting back the value of a , we can write the expression as :
$I = {\sin ^{ - 1}}\dfrac{x}{{\sqrt 7 }} + c$
Hence the value of the given integral expression $\int {\dfrac{1}{{\sqrt {7 - {x^2}} }}dx}$ is ${\sin ^{ - 1}}\dfrac{x}{{\sqrt 7 }} + c$ .
Thus, the correct option is option $\left( B \right)$.

Note:
As the question was of indefinite integral, we added an integral constant to the integration had it been the question of definite integral, we don’t have added an integral constant .
We can solve this question by directly using the formula of integration of the expression .
The general formula of integration is given as:
$\int {\dfrac{1}{{\sqrt {{a^2} - {x^2}} }}} dx = {\sin ^{ - 1}}\dfrac{x}{a} + c$