Indefinite Integrals Class 12 OP Malhotra Exe-15D ISC Maths Solutions Ch-15 Solutions. In this article you would learn about the integration of irrational functions i.e, the integrals of ∫dx/√a²-x² , ∫dx/√x²±a² . Step by step solutions of latest textbook has been given as latest syllabus. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

Indefinite Integrals Class 12 OP Malhotra Exe-15D ISC Maths Solutions Ch-15

Board	ISC
Publications	S Chand
Subject	Maths
Class	12th
Chapter-15	Indefinite Integrals
Writer	OP Malhotra
Exe-15(d)	integration of irrational functions i.e, the integrals of ∫dx/√a²-x²

Integration of irrational functions i.e, the integrals of ∫dx/√a²-x²

 Indefinite Integrals Class 12 OP Malhotra Exe-15D Solutions

Que-1: ∫dx/√1−x²

Sol: Let I = ∫dx/√1−x²
Put x= sin θ ⇒ θ = sin^-1x
⇒ dx = cos θ dθ
= ∫cosθ/cosθ dθ = θ + C = sin^-1x+ C

Que-2- ∫dx/√16−25x²

Sol: Let I = ∫dx/√16−25x² = 1/5∫dx/√16/25−x²
= 1/5∫dx/√(4/5)2−x²
= 1/5 sin^-1(x/4/5) + C
= 1/5 sin^-1(5x/4) + C

Que-3: ∫dx/√4x²+9

Sol: Let I = ∫dx/√4x²+9 = 1/2∫dx/√x²+(3/2)²
= 1/2 log∣x+√(x²+(3/2)²)∣+C
[∵ ∫dx/√x²+a² = log|x + √(x²+a²)| + C]

Que-4: ∫dx/√9x²−25

Sol: Let I = ∫dx/√9x²−25
= 1/3∫2x/√x²−(5/3)²
= 1/3log|x + √x²−(5/3)²| + C
[∵ ∫dx/√x²−a² = log | x + √x²−a²| + C ]

Que-5: ∫dx/√a²−b²x²

Sol: Let I = ∫dx/√a²−b²x² = 1/b∫dx/√(a/b)²−x²
= 1/b sin-1(x/a/b) + C
[∵ ∫dx/√a²−x² = sin^-1 x/a + C]
= 1/b sin^-1(bx/a) + C

Que-6: ∫dx/√(2−x)²+1

Sol: Let I = ∫dx/√(2−x)²+1;
Put 2 – x = t ⇒ -dx = dt
= ∫−dt/√t²+1² = -log|t + √(t²+1)| + C
= – log | 2 – x + √(2−x)²+1| + C

Que-7: ∫x+2/√x²+9 dx

Sol: Let I =∫x+2/√x²+9 dx
= ∫x/√x²+9 dx + 2∫dx/√x²+9
= 1/2∫(x²+9)^-1/2 2xdx + 2∫dx/√x²+3²

Que-8: ∫e^x/4−e^2x dx

Sol: Let I = ∫e^x/4−e^2x dx;
Put e^x = t ⇒ e^x dx = dt
= ∫dt/√4−t² = ∫dt/√2²−t²
= sin^-1(t/2) + C
[∵ ∫dx/√a²−x² = sin^-1(x/a) + C]
= sin^-1 (e^x/2) + C

Que-9: ∫x²/√x⁶−a⁶ dx

Sol: Let I = ∫x²/√x⁶−a⁶ dx
Put x³ = t ⇒ 3x²dx = dt
= ∫dt/3√t²−(a³)²
= 1/3log |t +√ t²−a⁶| + C
= 1/3log |x³ +√ x⁶−a⁶| + C

Que-10: ∫sec²x/tan²x+4 dx

Sol: Let I =∫sec²x/tan²x+4 dx
Put tan x = t ⇒ sec²x dx = dt
= ∫dt/t²+4 = ∫dt/t²+2² = 1/2 tan^-1(t/2) + C
[∵∫dx/x²+a² = 1/a tan^-1(x/a) + C]
= 1/2 tan^-1(tanx/2) + C
Or
Let I = ∫√sec²xtan²x+4 dx
Put tan x = t ⇒ sec²x dx = dt
= ∫dt/√t²+2² =log∣t+√t²+4∣+C
= log∣tanx+√tan²x+4∣+C

Que-11: ∫a^x/√1−a^x dx

Sol: Let I = ∫a^x/√1−a^x dx
Put a^x = t ⇒ a^x log a dx = dt
= ∫dt/loga √1-t² =1/loga sin^-1t+C
= 1/loga sin^-1(ax) + C

Que-12: ∫√1−x/1+x dx

Sol: Let I = ∫√1−x/1+x dx
= ∫√1−x/1+x×√1−x/1−x dx
= ∫1−x/√1−x² dx = ∫1/√1−x² dx−∫x/√1−x²
= sin^-1x + 1/2∫(1-x²)^-1/2 (−2x)dx
=sin^-1x + 1/2 (1-x²)^-1/2+1/-1/2+1 + C
= sin^-1x + √1−x² + C

Que-13: ∫(x−1)/√x²−1 dx

Sol: Let I = ∫(x−1)/√x²−1 dx
= ∫x/√x²−1 dx – ∫dx/√x²−1
Put x² – 1 = t ⇒ 2x dx = dt
= ∫dt/2√t – log|x + √x²−1| + C
= 1/2 t^-1/2+1/-1/2+1  – log|x + √x²-1| + C
= √t – log|x +√ x²−1| + C
= √x²−1 – log|x + √x²−1| + C

–: End of Indefinite Integrals Class 12 OP Malhotra Exe-15D ISC Math Ch-15 Solution :–

Return to :- OP Malhotra ISC Class-12 S Chand Publication Maths Solutions

Please share with your friends
Thanks