Integrate the function: \(\frac {sec^2x}{\sqrt {tan^2 x+4}}\)
Solution and Explanation
\(Let\ tan\ x = t\)
\(∴ sec^2x\ dx = dt\)
\(⇒ ∫\frac {sec^2x}{\sqrt {tan^2 x+4}}\ dx = ∫\frac {dt}{\sqrt {t^2+2^2}}\)
\(=log\ |t+\sqrt {t^2+4}|+C\)
\(=log\ |tan x+\sqrt {tan^2 x+4}|+C\)
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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.
