Let $l_n = \int \tan^{n} x \, dx , (n > 1) . l_4 + l_6 = a \, \, \tan^5 \, x + bx^5 + C$, where $C$ is a constant of integration, then the ordered pair $(a, b)$ is equal to :
- $\left(\frac{1}{5} , 0\right) $
- $\left(\frac{1}{5} , - 1 \right) $
- $\left( - \frac{1}{5} , 0\right) $
- $\left( - \frac{1}{5} , 1 \right) $
The Correct Option is A
Solution and Explanation
$l_{4}+l_{6}=\int\left(\tan ^{4} x+\tan ^{6} x\right) d x$
$=\int \tan ^{4} x \sec ^{2} x d x$
Let $\tan x=t$
$\sec ^{2} x d x=d t$
$=\int t^{4} d t$
$=\frac{t^{5}}{5}+C$
$=\frac{1}{5} \tan ^{5} x +C$
$a=\frac{1}{5},\, b=0$
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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.
