Integrate the function: \(\sqrt{1+\frac{x^2}{9}}\)
Solution and Explanation
Let \(I= \int \sqrt{1+\frac{x^2}{9}}dx=\frac{1}{3}\int \sqrt{9+x^2}dx=\frac{1}{3}\int\sqrt{(3)^2+x^2}\,dx\)
It is known that,\(\int \sqrt{x^2+a^2}dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\log\mid x+\sqrt{x^2+a^2}\mid+ C\)
∴\(I=\frac{1}{3}\bigg[\frac{x}{2}\sqrt{x^2+9}+\frac{9}{2}\log\mid x+\sqrt{x^2+9}\mid\bigg]+C\)
=\(\frac{x}{6}\sqrt{x^2+9}+\frac{3}{2}\log\mid x+\sqrt{x^2+9}\mid+C\)
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Concepts Used:
Integrals of Some Particular Functions
There are many important integration formulas which are applied to integrate many other standard integrals. In this article, we will take a look at the integrals of these particular functions and see how they are used in several other standard integrals.
Integrals of Some Particular Functions:
- ∫1/(x2 – a2) dx = (1/2a) log|(x – a)/(x + a)| + C
- ∫1/(a2 – x2) dx = (1/2a) log|(a + x)/(a – x)| + C
- ∫1/(x2 + a2) dx = (1/a) tan-1(x/a) + C
- ∫1/√(x2 – a2) dx = log|x + √(x2 – a2)| + C
- ∫1/√(a2 – x2) dx = sin-1(x/a) + C
- ∫1/√(x2 + a2) dx = log|x + √(x2 + a2)| + C
These are tabulated below along with the meaning of each part.
