Class 12 RD Sharma Solutions – Chapter 19 Indefinite Integrals – Exercise 19.1

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^{2log₃ x} dx

Solution:

∫ 3^{2log₃ x} dx = 

= ∫ x² dx

= x²⁺¹/(2+1) + Constant

= x³/3 + C

Question 2.  Evaluate

(i) 

Solution:

= 

We know, cos 2x = 2cos² x – 1

=

= ∫cos x dx

= sin x + Constant

(ii)

Solution:

= 

We know, cos 2x = 1 – 2sin² x

= 

= ∫ sin x dx

= -cos x + Constant

Question 3. Evaluate 

Solution:

dx 

= 

We know, e ^{log_e x} = x

= 

= ∫ x² dx

= x²⁺¹/2+1 + Constant

= x³/3 + C

Question 4. Evaluate: 

Solution:

 = ∫ 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) 

Solution:

= 

We know, cos 2x = 1 – 2sin² x

= ∫ 1/sin²x dx = ∫ cosec²x dx

= -cot x + Constant

(ii) 

Solution:

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