Question

Find the value of \[\int {\left\{ {\dfrac{{\left( {dx} \right)}}{{{x^2}\sqrt {4 - {x^2}} }}} \right\}} \].
A. \[\left( {1/4} \right)\left[ {\left( {\surd 4-{x^2}} \right)/x} \right] + c\]
B. \[\left( {1/2} \right)\left[ {\left( {\surd 4-{x^2}} \right)/x} \right] + c\]
C. \[\left( { - 1/4} \right)\left[ {\left( {\surd 4-{x^2}} \right)/x} \right] + c\]
D. \[\left( { - 1/2} \right)\left[ {\left( {\surd 4-{x^2}} \right)/x} \right] + c\]

Answer 1

Hint: In this question, we need to find the value of the given integration. For this, we have to simplify \[\sqrt {4 - {x^2}} \]as \[\sqrt {\dfrac{4}{{{x^2}}} - 1} \] and also put \[\dfrac{4}{{{x^2}}} - 1 = {t^2}\]. After that, we will differentiate this with respect to x and by putting the appropriate values in the given integration we can solve it.

Formula used: The following formula of integration is useful to solve the question.
\[\int {k{\text{ dx}} = {\text{ }}} kx + c\]
Where k is constant and c is the constant of integration.

Complete step-by-step solution:
Given that \[\int {\left\{ {\dfrac{{\left( {dx} \right)}}{{{x^2}\sqrt {4 - {x^2}} }}} \right\}} \]
Assume that \[I = \int {\left\{ {\dfrac{{\left( {dx} \right)}}{{{x^2}\sqrt {4 - {x^2}} }}} \right\}} \]
Let us simplify the above integration.
Now, we will take \[{x^2}\] common from the square root part.
Hence, we get
\[I = \int {\left\{ {\dfrac{{\left( {dx} \right)}}{{{x^3}\sqrt {\dfrac{4}{{{x^2}}} - 1} }}} \right\}} \]
Now, put \[\dfrac{4}{{{x^2}}} - 1 = {t^2}\]
By differentiating the above equation, we get
\[4\left( {\dfrac{{ - 1}}{{{x^4}}} \times 2x} \right)dx = 2tdt\]
\[\left( {\dfrac{{ - 4}}{{{x^3}}}} \right)dx = tdt\]
Now, put these values in the given integration.
So, we get
\[I = \int {\dfrac{{ - 1}}{4}\left\{ {\dfrac{{tdt}}{{\left( t \right)}}} \right\}} \]
\[I = \dfrac{{ - 1}}{4}\int {dt} \]
By integrating, we get
\[I = \dfrac{{ - 1}}{4}\left( t \right) + c\]
But \[\dfrac{4}{{{x^2}}} - 1 = {t^2}\]
That means, \[\sqrt {\dfrac{4}{{{x^2}}} - 1} = t\]
Thus, we get
\[I = \dfrac{{ - 1}}{4}\left( {\sqrt {\dfrac{4}{{{x^2}}} - 1} } \right) + c\]
Hence, the value of the integration \[\int {\left\{ {\dfrac{{\left( {dx} \right)}}{{{x^2}\sqrt {4 - {x^2}} }}} \right\}} \] is \[I = \dfrac{{ - 1}}{4}\left( {\sqrt {\dfrac{4}{{{x^2}}} - 1} } \right) + c\].

Therefore, the correct option is (C).


Additional information: The computation of an integral is known as integration. Integrals are often used in mathematics to obtain many useful quantities like areas, volumes, displacement, and so on. So if we talk about integrals, we generally mean definite integrals. Antiderivatives are represented by indefinite integrals. So, we can say that Integration in mathematics is a method of adding or summing the sections to obtain the whole. It is a reverse differentiation procedure in which we break down functions into parts.

Note: Many students generally make mistakes in taking \[{x^2}\] common from the square root part. That may get the wrong result after simplification. Also, students generally forgot to take the derivative of \[{x^2}\] while taking the derivative of \[\dfrac{4}{{{x^2}}} - 1 = {t^2}\].