Match the statements of Column I with values of Column II- Column I Column II- A -e2x-2ex-e2x-1dx - A ln e2x-1 - B tan-1ex-c p A - -1-2- B - -1-4- B -x-x2-2dx - A x-x2-23-2 - B-x-x2-2-c q A - 1-2- B - -2- C -cos 8x-cos 7x-1-2 cos 5xdx -A sin 3x-B sin 2x-c r A - 1-3- B-2- D -ln x-x3dx - A ln x-x2-B-x2-c s A - 1-3- B-1-2- -

(A) ∫e^2x/e^2x+1dx 2 ∫e^x/e^2x+1dx =1/2 ln (1+e^2x) 2 tan^ 1(e^x)+c (B) x+√(x^2+2)=t ⇒ x^2+2 = t^2+x^2 2tx ⇒ x = 1/2 ( t 2/t ) So, I = 1/2∫ t^1/2 ( 1+2/t^2 ) ...

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Byju's Answer

Standard XIII

Mathematics

Theorems on Integration

Match the sta...

Question

Match the statements of Column I with values of Column II
$\begin{matrix} C o l u m n I & C o l u m n I I (A) \int \frac{e^{2 x} - 2 e^{x}}{e^{2 x} + 1} d x = A l n (e^{2 x} + 1) + B t a n^{- 1} (e^{x}) + c & (p) A = - \frac{1}{2}, B = - \frac{1}{4} (B) \int \sqrt{x + \sqrt{x^{2} + 2}} d x = A {x + \sqrt{x^{2} + 2}}^{\frac{3}{2}} + \frac{B}{\sqrt{x + \sqrt{x^{2} + 2}}} + c & (q) A = \frac{1}{2}, B = - 2 (C) \int \frac{c o s 8 x - c o s 7 x}{1 + 2 c o s 5 x} d x = A s i n 3 x + B s i n 2 x + c & (r) A = \frac{1}{3}, B = - 2 (D) \int \frac{l n x}{x^{3}} d x = A \frac{l n x}{x^{2}} + \frac{B}{x^{2}} + c & (s) A = \frac{1}{3}, B = - \frac{1}{2} \end{matrix}$

A-s , B-r , C-q , D-p

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A-p , B-s , C-r , D-q

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A-p , B-q , C-r , D-s

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A-q , B-r , C-s , D-p

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Solution

The correct option is D
A-q , B-r , C-s , D-p

(A) $\int \frac{e^{2 x}}{e^{2 x} + 1} d x - 2 \int \frac{e^{x}}{e^{2 x} + 1} d x = \frac{1}{2} l n (1 + e^{2 x}) - 2 t a n^{- 1} (e^{x}) + c$

(B) $x + \sqrt{x^{2} + 2} = t$
$\Rightarrow x^{2} + 2 = t^{2} + x^{2} - 2 t x$
$\Rightarrow x = \frac{1}{2} (t - \frac{2}{t})$
So, $I = \frac{1}{2} \int t^{\frac{1}{2}} (1 + \frac{2}{t^{2}}) d t$
$I = \frac{1}{2} \int t^{\frac{1}{2}} d t + \int t^{- \frac{3}{2}} d t = \frac{1}{3} t^{\frac{3}{2}} - \frac{2}{\sqrt{t}} + c$

(C) $\int \frac{c o s 8 x - c o s 7 x}{1 + 2 c o s 5 x} d x$
$= \int \frac{- 2 s i n \frac{5 x}{2} s i n \frac{x}{2} (3 - 2 + 2 c o s 5 x) d x}{1 + 2 c o s 5 x}$ $[∵ s i n 3 (\frac{5 x}{2}) = 3 s i n \frac{5 x}{2} - 4 s i n^{3} \frac{5 x}{2}]$
$= \int (c o s 3 x - c o s 2 x) d x = \frac{s i n 3 x}{3} - \frac{s i n 2 x}{2} + c$

(D) $\int \frac{l n x}{x^{3}} d x = (l n x) (- \frac{1}{2 x^{2}}) + \int \frac{1}{x} . \frac{1}{2 x^{2}} d x$
$= (- \frac{1}{2}) \frac{l n x}{x^{2}} + (- \frac{1}{4}) \frac{1}{x^{2}} + c$

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Q. Match the entries of Column - I and Column - II.

	Column - I		Column - II
a	If 4 $s i n^{- 1} x + c o s^{- 1} x = π$ , then x equals	1	ab
b	If $∠ C = 90^{0}$ , then the value of $t a n^{- 1}$ $\frac{a}{b + c}$ + $t a n^{- 1}$ $\frac{b}{c + a}$ is	2	$π$
c	$t a n^{- 1}$ 1 + $t a n^{- 1}$ 2 + $t a n^{- 1}$ 3 is	3	$π$ /4
d	If $s e c^{- 1}$ $\frac{x}{a}$ - $s e c^{- 1}$ $\frac{x}{b}$ = $s e c^{- 1}$ b - $s e c^{- 1}$ a, then x equals	4	1/2

Q. Match the following

	List I		List II
A.	$c o s (3 c o s^{- 1} x)$	1.	$2 x \sqrt{1 - x^{2}}$
B.	$s i n (2 s i n^{- 1} x)$	2.	$4 x^{3} - 3 x$
C.	$t a n (3 t a n^{- 1} x)$	3.	$\frac{3 x - x^{3}}{1 - 3 x^{2}}$
D.	$s i n (2 c o s^{- 1} x)$	4.	$2 x^{2} - 1$

Match the statements of Column I with values of Column II
$\begin{matrix} C o l u m n I & C o l u m n I I (A) \int \frac{e^{2 x} - 2 e^{x}}{e^{2 x} + 1} d x = A l n (e^{2 x} + 1) + B t a n^{- 1} (e^{x}) + c & (p) A = - \frac{1}{2}, B = - \frac{1}{4} (B) \int \sqrt{x + \sqrt{x^{2} + 2}} d x = A {x + \sqrt{x^{2} + 2}}^{\frac{3}{2}} + \frac{B}{\sqrt{x + \sqrt{x^{2} + 2}}} + c & (q) A = \frac{1}{2}, B = - 2 (C) \int \frac{c o s 8 x - c o s 7 x}{1 + 2 c o s 5 x} d x = A s i n 3 x + B s i n 2 x + c & (r) A = \frac{1}{3}, B = - 2 (D) \int \frac{l n x}{x^{3}} d x = A \frac{l n x}{x^{2}} + \frac{B}{x^{2}} + c & (s) A = \frac{1}{3}, B = - \frac{1}{2} \end{matrix}$

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