Integrate the function: \(\frac {1}{\sqrt {9-25x^2}}\)
Solution and Explanation
\(Let \ 5x = t\)
\(∴ 5 dx = dt\)
⇒ \(∫\frac {1}{\sqrt {9-25x^2}} \ dx\) \(=\) \(\frac {1}{5} ∫\frac {1}{9-t^2} dt\)
\(=\frac 15 ∫\frac {1}{\sqrt {3^2-t^2}} \ dt\)
\(=\frac 15 sin^{-1}(\frac t3)+C\)
\(=\frac 15 sin^{-1}(\frac {5x}{3})+C\)
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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.
