# Integration of Square Root (a^2-x^2) | Integral of Root (a^2-x^2)

In this post, we will find the integral of root(a2-x2). The integration of square root of a2-x2 is given as follows

∫ √(a2-x2) dx = x/2 ⋅ √(a2-x2) + a2/2 sin-1(x/a) + C, where C is an integration constant.

## Integration of $\sqrt{a^2-x^2}$

Let $I = \int \sqrt{a^2-x^2} dx$ $\cdots (I)$

We will find the integration of the square root of a2-x2 using the integration by parts formula. The formula says that if f(x), g(x) are two functions then the integration of f(x)g(x) is given by

$\int f(x)g(x) dx$ $=f \int g dx$ $-\int \big[\dfrac{df}{dx} \int g dx \big]dx$ $\cdots (\star)$

Put $f(x)=\sqrt{a^2-x^2}$ and $g(x)=1$.

Now,

and

Now, using the formula $(\star)$, the integral of root(a^2-x^2) is

I = $\int \sqrt{a^2-x^2} dx = \int \sqrt{a^2-x^2} \cdot 1 dx$

= $\sqrt{a^2-x^2} \cdot x$ $-\int \dfrac{-x}{\sqrt{a^2-x^2}} \cdot x dx$

= $x\sqrt{a^2-x^2}$ $-\int \dfrac{a^2-x^2-a^2}{\sqrt{a^2-x^2}}$

= $x\sqrt{a^2-x^2}-\int \sqrt{a^2-x^2} dx$ $+a^2 \int \dfrac{dx}{a^2-x^2}$

Therefore,

I = $x\sqrt{a^2-x^2}-I$ $+a^2 \int \dfrac{dx}{a^2-x^2}$

⇒ 2I = $x\sqrt{a^2-x^2}$ $+a^2 \int \dfrac{dx}{a^2-x^2}$

⇒ 2I = $x\sqrt{a^2-x^2}$ $+a^2 \sin^{-1} \dfrac{x}{a}+C$

∴ The integral of root a2-x2 is equal to $\dfrac{x}{2} \sqrt{a^2-x^2}$ $+\dfrac{a^2}{2}\sin^{-1} \dfrac{x}{a}+C$, where C is an integration constant.

## Application

Using the above formula, we can easily find the integral of square root of 4-x2.

Put a=2 in the above formula.

So the integration of root 4-x2 is equal to

∫ $\sqrt{4-x^2} dx$ $=\dfrac{x}{2} \sqrt{4-x^2}$ +2 sin-1(x/2) +C.

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## FAQs

Q1: What is the integration of square root of a2-x2?

Answer: The integration of square root of a2-x2 is equal to ∫√(a2-x2) dx = x/2 ⋅ √(a2-x2) + a2/2 sin-1(x/a)+C where C is an integration constant.