RD Sharma Class 12 Ex 19.1 Solutions Chapter 19 Indefinite Integrals

Here we provide RD Sharma Class 12 Ex 19.1 Solutions Chapter 19 Indefinite Integrals for English medium students, Which will very helpful for every student in their exams. Students can download the latest Sharma Class 12 Ex 19.1 Solutions Chapter 19 Indefinite Integrals book pdf download. Now you will get step-by-step solutions to each question.

TextbookNCERT
ClassClass 12th
SubjectMaths
Chapter19
Exercise19.1
CategoryRD Sharma Solutions

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) ∫ x5/4 dx

Solution: 

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

= 4/9 x9/4 + 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/x3/2 dx

Solution: 

∫ x-3/2 dx = x-3/2 + 1/(-3/2 +1) + Constant

= x-1/2/(-1/2) + C

= -2/(√x)+ C

(v) ∫ 3x dx

Solution: 

∫ 3x dx = 3x/log3 + Constant

(vi) ∫ 1/x2/3 dx

Solution: 

∫ 1/x2/3 dx = ∫ x-2/3 dx

= x-2/3 + 1/(-2/3+1) + Constant

= x1/3/(1/3) + C

= 3x1/3 + C

(vii) ∫ 32log3 x dx

Solution: 

∫ 32log3 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 loge 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/loge (ab)-1 + Constant

= -a-x b-x/loge (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/cos2 x dx = ∫ sec2 x dx

= tan x + Constant

Question 6. Evaluate: ∫ elog√x /x  dx

Solution:  

∫ eloge √x /x dx = ∫√x/x dx

= ∫ x-1/2 dx = x-1/2 + 1/(-1/2 + 1) + Constant

= x1/2 /(1/2) + C

= 2√x + C

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