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Question: Integrate the function \[\sqrt {4 - {x^2}} \]...

Integrate the function 4−x2\sqrt {4 - {x^2}}

Explanation

Solution

Hint : The simple meaning of trigonometry is calculations of triangles. For solving this question, we will use some identities, As there is a square root present in the question. We have some formulas/ identities, so using that question will be very easy to solve.
(A) ∫a2−x2dx=x2a2−x2+a22sin⁡−1xa+c\int {\sqrt {{a^2} - {x^2}} } dx = \dfrac{x}{2}\sqrt {{a^2} - {x^2}} + \dfrac{{{a^2}}}{2}{\sin ^{ - 1}}\dfrac{x}{a} + c
By taking the above identity, we can solve the given question.

Complete step-by-step answer :
We also have some identities
(A) ∫a2−x2dx=x2a2−x2+a22sin⁡−1xa+c\int {\sqrt {{a^2} - {x^2}} } dx = \dfrac{x}{2}\sqrt {{a^2} - {x^2}} + \dfrac{{{a^2}}}{2}{\sin ^{ - 1}}\dfrac{x}{a} + c
(B) ∫a2+x2dx=x2a2+x2+a22log⁡∣x+x2+a2∣+c\int {\sqrt {{a^2} + {x^2}} } dx = \dfrac{x}{2}\sqrt {{a^2} + {x^2}} + \dfrac{{{a^2}}}{2}\log \left| {x + \sqrt {{x^2} + {a^2}} } \right| + c
(C) ∫x2−a2dx=x2x2−a2−a22log⁡∣x+x2−a2∣+c\int {\sqrt {{x^2} - {a^2}} } dx = \dfrac{x}{2}\sqrt {{x^2} - {a^2}} - \dfrac{{{a^2}}}{2}\log \left| {x + \sqrt {{x^2} - {a^2}} } \right| + c
For solving this question, we will use the identity (A),
According to that the value of a is 2 and x is x, so-
∫a2−x2dx=x2a2−x2+a22sin⁡−1xa+c\int {\sqrt {{a^2} - {x^2}} } dx = \dfrac{x}{2}\sqrt {{a^2} - {x^2}} + \dfrac{{{a^2}}}{2}{\sin ^{ - 1}}\dfrac{x}{a} + c
Now, use this identity,
In question, Given that, ∫4−x2\int {\sqrt {4 - {x^2}} } , for integrating the given function, we can use the identity.
Here, 4=(2)24 = {(2)^2} .
So = ∫(2)2−x2dx\int {\sqrt {{{\left( 2 \right)}^2} - {x^2}} } dx (Start substituting in the given identity)
We get,= x24−x2+42sin⁡−1x2+c\dfrac{x}{2}\sqrt {4 - {x^2}} + \dfrac{4}{2}{\sin ^{ - 1}}\dfrac{x}{2} + c
= x4−x22+42sin⁡−1(x2)+c\dfrac{{x\sqrt {4 - {x^2}} }}{2} + \dfrac{4}{2}{\sin ^{ - 1}}\left( {\dfrac{x}{2}} \right) + c
∫4−x2\int {\sqrt {4 - {x^2}} } = x4−x22+42sin⁡−1(x2)+c\dfrac{{x\sqrt {4 - {x^2}} }}{2} + \dfrac{4}{2}{\sin ^{ - 1}}\left( {\dfrac{x}{2}} \right) + c
So, this will be the answer.
So, the correct answer is “ x4−x22+42sin⁡−1(x2)+c\dfrac{{x\sqrt {4 - {x^2}} }}{2} + \dfrac{4}{2}{\sin ^{ - 1}}\left( {\dfrac{x}{2}} \right) + c ”.

Note : In this question, we are using identities, by using that question will be easy to solve. By choosing the correct identity for solving the question remember to check the a and x value. Also remember to check the positive and negative signs. The simple meaning of trigonometry is calculations of triangles. Also, in physics, trigonometry is used to find the components of vectors and also in projectile motion have a lot of application of trigonometry.