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\begin{align} \int \left(\sqrt{x} - \frac {1}{\sqrt{x}}\right)^2 .dx\e...

egin{align} int left(sqrt{x} - fra | Class 12 Mathematics Chapter Integrals, Integrals NCERT Solutions

Question: \begin{align} \int \left(\sqrt{x} - \frac {1}{\sqrt{x}}\right)^2 .dx\end{align}

Answer:

\begin{align} \int \left(\sqrt{x} - \frac {1}{\sqrt{x}}\right)^2 .dx\end{align}

\begin{align} =\int \left(x + \frac {1}{x} - 2\right) .dx\end{align}

\begin{align} =\int x .dx + \int \frac {1}{x} . dx - 2\int 1. dx\end{align}

\begin{align} =\frac{x^2}{2} + \log\left(\left|x\right|\right) - 2x + C\end{align}

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Popular questions of class 12 Mathematics

Q: Given an example of a relation. Which is
(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Q: Determine whether each of the following relations are reflexive, symmetric and transitive:
(i) Relation R in the set A = {1, 2, 3,13, 14} defined as
R = {(x, y): 3x − y = 0}
(ii) Relation R in the set N of natural numbers defined as
R = {(x, y): y = x + 5 and x < 4}
(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as
R = {(x, y): y is divisible by x}
(iv) Relation R in the set Z of all integers defined as
R = {(x, y): x − y is as integer}
(v) Relation R in the set A of human beings in a town at a particular time given by
(a) R = {(x, y): x and y work at the same place}
(b) R = {(x, y): x and y live in the same locality}
(c) R = {(x, y): x is exactly 7 cm taller than y}
(d) R = {(x, y): x is wife of y}
(e) R = {(x, y): x is father of y}

Q: Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Q: 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.

Q: 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.

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

(i) f : R → R defined by f(x) = 3 – 4x

(ii) f : R → R defined by f(x) = 1 + x²

Q: Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.

Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

Q: Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.

Recently Viewed Questions of Class 12 Mathematics

Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.

Determine order and degree(if defined) of differential equation yⁿ + (y')² + 2y =0

Determine order and degree(if defined) of differential equation (y^m)² + (yⁿ)³ + (y')⁴ + y⁵ =0

y = e^x +1 : yⁿ -y^' = 0

Q: Show that the relation R in the set A of points in a plane given by R = {(P, Q): distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all point related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.

Q: If a matrix has 24 elements, what are the possible order it can have? What, if it has 13 elements?

Q: \begin{align} \int\left(2x^2-3Sinx +5\sqrt {x}\right).dx\end{align}

Q: Show that the relation R defined in the set A of all polygons as R = {(P1, P2): P1 and P2 have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?

Let f, g and h be functions from R to R. Show that

(f + g)oh = foh + goh

(f . g)oh = (foh) . (goh)

Q: Show that each of the relation R in the set A = { x ∈Z: 0≤x≤12}, A={x} given by
(i) R = { (a,b) : |a - b| is a multiple of 4}
(ii) R = {(a,b):a = b} is an equivalence relation.
Find the set of all elements related to 1 in each case.

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Welcome to the NCERT Solutions for Class 12 Mathematics - Chapter . This page offers a step-by-step solution to the specific question from Excercise 1 , Question 10: egin{align} int left(sqrt{x} - frac {1}{sqrt{x}} ight)^2 .dxend{align}....

egin{align} int left(sqrt{x} - fra | Class 12 Mathematics Chapter Integrals, Integrals NCERT Solutions

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egin{align} int left(sqrt{x} - fra | Class 12 Mathematics Chapter Integrals, Integrals NCERT Solutions