RD Sharma Class 12 Ex 19.1 Solutions Chapter 19 Indefinite Integrals

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Textbook	NCERT
Class	Class 12th
Subject	Maths
Chapter	19
Exercise	19.1
Category	RD Sharma Solutions

Table of Contents

RD Sharma Class 12 Ex 19.1 Solutions Chapter 19 Indefinite Integrals

Question 1. Integrate the following integrals with respect to x:

(i) ∫ x⁴ dx

Solution:

∫ x⁴ dx = x⁴⁺¹/(4+1) + Constant
= x⁵/5 + C

(ii) ∫ x^5/4 dx

Solution:

∫ x^5/4 dx = x^{5/4 + 1}/(5/4 +1) + Constant
= 4/9 x^9/4 + C

(iii) ∫ 1/x⁵ dx

Solution:

∫ 1/x⁵ dx = ∫ x^-5 dx
= x^-5+1/(-5+1) + Constant
= x^-4/(-4)+ C
= -1/(4x⁴) + C

(iv) ∫ 1/x^3/2 dx

Solution:

∫ x^-3/2 dx = x^{-3/2 + 1}/(-3/2 +1) + Constant
= x^-1/2/(-1/2) + C
= -2/(√x)+ C

(v) ∫ 3^x dx

Solution:

∫ 3^x dx = 3^x/log3 + Constant

(vi) ∫ 1/x^2/3 dx

Solution:

∫ 1/x^2/3 dx = ∫ x^-2/3 dx
= x^{-2/3 + 1}/(-2/3+1) + Constant
= x^1/3/(1/3) + C
= 3x^1/3 + C

(vii) ∫ 3^2log_³^x dx

Solution:

∫ 3^2log_³^x dx = $∫ 3^{log_3 x^2} dx$
= ∫ x² dx
= x²⁺¹/(2+1) + Constant
= x³/3 + C

Question 2. Evaluate

(i) $∫\sqrt{\frac{(1 + cos 2x)}{2} }dx$

Solution:

$∫\sqrt{\frac{(1 + cos 2x)}{2} }dx$

= $∫\sqrt{\frac{(1 + 2cos^2x - 1)}{2}} dx$

We know, cos 2x = 2cos² x – 1

= $∫\sqrt{\frac{(2cos^2x)}{2}} dx$

= ∫cos x dx

= sin x + Constant

(ii) $∫\sqrt{\frac{(1 - cos 2x)}{2} }dx$

Solution:

$∫\sqrt{\frac{(1 - cos 2x)}{2} }dx$

= $∫\sqrt{\frac{(1 - 1 + 2sin^2x)}{2}} dx$

We know, cos 2x = 1 – 2sin² x

= $∫\sqrt{\frac{(2sin^2 x)}{2}} dx$

= ∫ sin x dx

= -cos x + Constant

Question 3. Evaluate $∫ \frac{e^{6log_ex}-e^{5log_ex}}{e^{4log_ex}-e^{3log_ex}}$

Solution:

$∫ \frac{e^{6log_ex}-e^{5log_ex}}{e^{4log_ex}-e^{3log_ex}}$ dx
= $∫\frac{(x^6 - x^5)}{(x^4 - x^3)} dx$
We know, e ^log_^e^x = x
= $∫ \frac{x^5(x - 1)}{x^3(x - 1)} dx$
= ∫ x² dx
= x²⁺¹/2+1 + Constant
= x³/3 + C

Question 4. Evaluate: $∫ \frac{1}{a^x b^x}dx$

Solution:

$∫ \frac{1}{a^x b^x}dx$ = ∫ a^-x b^-x dx
= ∫ (ab)^-x dx
= (ab)^-x/log_e (ab)^-1 + Constant
= -a^-x b^-x/log_e (ab) + C
or
= -a^-x b^-x/ ln(ab) + C

Question 5. Evaluate

(i) $∫ \frac{cos 2x + 2sin^2 x}{sin^2 x} dx$

Solution:

$∫ \frac{cos 2x + 2sin^2 x}{sin^2 x} dx$

= $∫ \frac{1 - 2sin^2x + 2sin^2x}{sin^2x} dx$

We know, cos 2x = 1 – 2sin² x

= ∫ 1/sin²x dx = ∫ cosec²x dx

= -cot x + Constant

(ii) $∫\frac{2cos^2x - (cos2x)}{cos^2x}dx$

Solution:

$∫\frac{2cos^2x - (cos2x)}{cos^2x}dx$

$∫\frac{2cos^2x - (2cos^2x -1)}{cos^2x}dx$

We know, cos 2x = 2cos² x – 1

= ∫ 1/cos²x dx = ∫ sec² x dx

= tan x + Constant

Question 6. Evaluate: ∫ e^log√x /x dx

Solution:

∫ e^log_^e^√x /x dx = ∫√x/x dx
= ∫ x^-1/2 dx = x^{-1/2 + 1}/(-1/2 + 1) + Constant
= x^1/2 /(1/2) + C
= 2√x + C

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RD Sharma Class 12 Ex 19.1 Solutions Chapter 19 Indefinite Integrals

Question 1. Integrate the following integrals with respect to x:

(i) ∫ x4 dx

(ii) ∫ x5/4 dx

(iii) ∫ 1/x5 dx

(iv) ∫ 1/x3/2 dx

(v) ∫ 3x dx

(vi) ∫ 1/x2/3 dx

(vii) ∫ 32log3 x dx

Question 2. Evaluate

(i)

(ii)

Question 3. Evaluate

Question 4. Evaluate:

Question 5. Evaluate

(i)

(ii)

Question 6. Evaluate: ∫ elog√x /x dx

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(i) ∫ x⁴ dx

(ii) ∫ x^5/4 dx

(iii) ∫ 1/x⁵ dx

(iv) ∫ 1/x^3/2 dx

(v) ∫ 3^x dx

(vi) ∫ 1/x^2/3 dx

(vii) ∫ 3^2log_³^x dx

(i) $∫\sqrt{\frac{(1 + cos 2x)}{2} }dx$

(ii) $∫\sqrt{\frac{(1 - cos 2x)}{2} }dx$

Question 3. Evaluate $∫ \frac{e^{6log_ex}-e^{5log_ex}}{e^{4log_ex}-e^{3log_ex}}$

Question 4. Evaluate: $∫ \frac{1}{a^x b^x}dx$

(i) $∫ \frac{cos 2x + 2sin^2 x}{sin^2 x} dx$

(ii) $∫\frac{2cos^2x - (cos2x)}{cos^2x}dx$

Question 6. Evaluate: ∫ e^log√x /x dx