Question:
Integrate $ \frac{{{\sec }^{2}}\,({{\sin }^{-1}}x)}{\sqrt{1-{{x}^{2}}}} $
Integrate $ \frac{{{\sec }^{2}}\,({{\sin }^{-1}}x)}{\sqrt{1-{{x}^{2}}}} $
Updated On: Jun 23, 2024
- $ \sin \,({{\tan }^{-1}}x)+c$
- $ \tan \,({{\sec }^{-1}}x)+c $
- $ \tan \,({{\sin }^{-1}}x)+c $
- $ -\tan \,(co{{s}^{-1}}x)+c $
Hide Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Let $ l=\int{\frac{{{\sec }^{2}}({{\sin }^{-1}}x)}{\sqrt{1-{{x}^{2}}}}}dx $ Again, let $ {{\sin }^{-1}}x=t $
$ \Rightarrow $ $ \frac{dt}{dx}=\frac{1}{\sqrt{1-{{x}^{2}}}} $
$ \Rightarrow $ $ dt=\frac{1}{\sqrt{1-{{x}^{2}}}}\,dx $
$ \therefore $ $ l=\int{{{\sec }^{2}}\,t\,dt} $
$=\tan \,t+C $
$=\tan ({{\sin }^{-1}}x)+C $
$ \Rightarrow $ $ \frac{dt}{dx}=\frac{1}{\sqrt{1-{{x}^{2}}}} $
$ \Rightarrow $ $ dt=\frac{1}{\sqrt{1-{{x}^{2}}}}\,dx $
$ \therefore $ $ l=\int{{{\sec }^{2}}\,t\,dt} $
$=\tan \,t+C $
$=\tan ({{\sin }^{-1}}x)+C $
Was this answer helpful?
0
0
Top Questions on Integrals of Some Particular Functions
- If \(\int \frac{dx}{(x+1)(x-2)(x-3)}=\frac{1}{k}log_e\left \{ \frac{|x-3|^3|x+1|}{(x-2)^4}\right \}+c\), then the value of k is
- WBJEE - 2023
- Mathematics
- Integrals of Some Particular Functions
- Let $f(x)=\int \frac{2 x}{\left(x^2+1\right)\left(x^2+3\right)} d x$ If $f(3)=\frac{1}{2}\left(\log _e 5-\log _e 6\right)$, then $f(4)$ is equal to
- JEE Main - 2023
- Mathematics
- Integrals of Some Particular Functions
- If $\int \sqrt{\sec 2 x-1} d x=\alpha \log _e\left|\cos 2 x+\beta+\sqrt{\cos 2 x\left(1+\cos \frac{1}{\beta} x\right)}\right|+$ constant, then $\beta-\alpha$ is equal to_______
- JEE Main - 2023
- Mathematics
- Integrals of Some Particular Functions
- If $\int \limits_0^\pi \frac{5^{\cos x}\left(1+\cos x \cos 3 x+\cos ^2 x+\cos ^3 x \cos 3 x\right) d x}{1+5^{\cos x}}=\frac{ k \pi}{16}$, then $k$ is equal to
- JEE Main - 2023
- Mathematics
- Integrals of Some Particular Functions
- The integral $16 \int\limits_1^2 \frac{d x}{x^3\left(x^2+2\right)^2}$ is equal to
- JEE Main - 2023
- Mathematics
- Integrals of Some Particular Functions
View More Questions
Questions Asked in JKCET exam
- A moving block having mass $m$, collides with another stationary block having mass $4\,m$. The lighter block comes to rest after collision. When the initial velocity of the lighter block is $v$, then the value of coefficient of restitution (e) will be
- JKCET - 2019
- work, energy and power
- A force $ F = 3x^2 + 2x + 1 $ acts on a body in the $ x $ -direction. The work done by this force during a displacement from $ x = -1 $ to $ +1 $ is
- JKCET - 2019
- work, energy and power
- What is the colour of the flame on heating potassium in the flame of a Bunsen burner?
- JKCET - 2019
- GROUP 1 ELEMENTS
- If $ a $ , $ b $ , $ c $ are the lengths of three edges of unit cell, $ \alpha $ is the angle between side $ b $ and $ c $ , $ \beta $ is the angle between side $ a $ and $ c $ and $ \gamma $ is the angle between side $ a $ and $ b $ , what is the Bravais Lattice dimensions of a tetragonal crystal?
- JKCET - 2019
- The solid state
- Which of the following compounds is known as copper glance?
- JKCET - 2019
- General Principles and Processes of Isolation of Elements
View More Questions
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.
