\(∫\frac{dx}{\sqrt{9x-4x^2}}\) equals
\(\frac{1}{9}sin^{-1}(\frac{9x-8}{8})+C\)
\(\frac{1}{2}sin^{-1}(\frac{8x-9}{9})+C\)
\(\frac{1}{3}sin^{-1}(\frac{9x-8}{8})+C\)
\(\frac{1}{2}sin^{-1}(\frac{9x-8}{9})+C\)
The Correct Option is B
Solution and Explanation
\(∫\frac{dx}{\sqrt{9x-4x^2}}\)
\(= ∫\frac{1}{\sqrt{-4(x^2-\frac{9}{4x})}}dx\)
\(= ∫\frac{1}{-4(x^2-\frac{9}{4}x+\frac{81}{64}-\frac{81}{64})}dx\)
\(= ∫\frac{1}{\sqrt{-4[(x-\frac{9}{8})^2-(\frac{9}{8})^2}}] dx\)
\(=\frac{1}{2} ∫\frac{1}{\sqrt{(\frac{9}{8})^2 - (x-\frac{9}{8})^2}} dx\)
\(=\frac{1}{2}[sin^{-1}\bigg(\frac{x-\frac{9}{8}}{\frac{9}{8}}\bigg)]+C\) \(( ∫\frac{dy}{\sqrt{a^2-y^2}} = sin^{-1} \frac{y}{a}+C)\)
\(= \frac{1}{2}sin^{-1}(\frac{8x-9}{9})+C\)
Hence, the correct Answer is B.
Top CBSE CLASS XII Mathematics Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A committee of 11 members is to be formed from 8 males and 5 females. Let m denotes the number of ways of selecting committee having atleast 6 males and n denotes the number of ways of selecting atleast 3 females then
Determine whether each of the following relations are reflexive, symmetric, and transitive.
- Relation R in the set A={1,2,3,...13,14} defined as R={(x,y): 3x-y=0}.
- Relation R in the set of N natural numbers defined as R={(x,y): y=x+5 and x<4}.
- Relation R in the set A={1,2,3,4,5,6} defined as R={(x,y): y is divisible by x}.
- Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
- Relation R in the set of human beings in a town at a particular time given by
- R={(x,y): x and y work at same place}
- R={(x,y): x and y live in the same locality}
- R={(x,y): x is exactly 7 am taller that y}
- R={(x,y): x is wife of y}
- R={(x,y):x is father of y}
Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
Top CBSE CLASS XII integral Questions
- Evaluate the integral: \(\int_{0}^{2}\frac{dx}{x+4-x^2}\)
Find the area of the region bounded by the curve y2=x and the lines x=1,x=4 and the x-axis
Find the area of the region bounded by y2=9x, x=2, x=4 and the x-axis in the first quadrant.
Find the area of the region bounded by x2=4y,y=2,y=4 and the x-axis in the first quadrant.
Find the area of the region bounded by the ellipse \(\frac{x^2}{16}+\frac{y^2}{9}=1\)
Top CBSE CLASS XII Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A boy of mass 50 kg is standing at one end of a, boat of length 9 m and mass 400 kg. He runs to the other, end. The distance through which the centre of mass of the boat boy system moves is
- A capillary tube of radius r is dipped inside a large vessel of water. The mass of water raised above water level is M. If the radius of capillary is doubled, the mass of water inside capillary will be
- A circular disc is rotating about its own axis. An external opposing torque 0.02 Nm is applied on the disc by which it comes rest in 5 seconds. The initial angular momentum of disc is
- A circular disc is rotating about its own axis at uniform angular velocity \(\omega.\) The disc is subjected to uniform angular retardation by which its angular velocity is decreased to \(\frac {\omega}{2}\) during 120 rotations. The number of rotations further made by it before coming to rest is
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.
