Indefinite Integrals Class 12 OP Malhotra Exe-13B ISC Maths Solutions Ch-13 Solutions. In this article you would learn about integrals of 1/x , 1/ax+b , e^x , a^x and e^ax . 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-13B ISC Maths Solutions Ch-13

Board	ISC
Publications	S Chand
Subject	Maths
Class	12th
Chapter-13	Indefinite Integrals
Writer	OP Malhotra
Exe-13(b)	integrals of 1/x , 1/ax+b , ex , ax and eax

Exercise- 13B

Indefinite Integrals Class 12 OP Malhotra Exe-13B Solution.

Que-1: ∫(x²+1)²/x³ dx

Sol:

Que-2: ∫x+1/x²-1 dx

Sol:

Que-3: ∫e^-x dx

Sol:

Que-4: ∫e^3x dx

Sol: ∫e^3x dx = e^3x/3 + C

Que-5: ∫a^2x dx

Sol:

Que-6: ∫(e^3alog_ex + e^3xlog_ea) dx

Sol:  Let I =∫(e^3alog_ex + e^3xlog_ea) dx
= ∫(e^{log_ex^(3a)} + e^{log_ea^(3x)}) dx
= ∫x^3a dx + ∫a^3x dx [∴e^logx = x]
= x^3a+1/3a+1 + a^3x/3loga + C

Que-7: ∫(3e^2x+ 3e^4x)/(e^x + e^-x) dx

Sol: ∫(3e^2x+ 3e^4x)/(e^x + e^-x) dx
= ∫3e^2x(1+e^2x )e^x/(e^2x+1) dx
= ∫3e^3x dx
= 3e^3x/3 + C = e^3x + C

Que-8: ∫(5x+7/x +e^x) dx

Sol: ∫(5x+7/x +e^x) dx
=∫[5 + 7/x + e^x] dx
= ∫5dx + 7∫1/x dx + ∫e^x dx
=5x + 7log|x| + e^x + C

Que-9: ∫(ax²+bx+c/x²) dx

Sol: Let I =∫(ax²+bx+c/x²) dx
= ∫[a + b/x + c/x²]dx
=ax + blog|x| – c/x + C

Que-10: ∫dx/√16-x² 

Sol: ∫dx/√16-x² = ∫dx/√4²-x²
= sin^-1x/4 + C

Que-11: ∫dx/√25-4x² 

Sol:

Que-12: ∫dx/√4+x² 

Sol: ∫dx/√4+x² = ∫dx/√2²+x²
= 1/2 tan^-1(x/2) + C

Que-13: ∫dx/√16+9x²

Sol:

Que-14: ∫(6/1+x² + 10^x – 5cosec²x) dx

Sol:

Que-15: ∫(x+1/x)(x²+1/x²) dx

Sol : ∫(x+1/x)(x²+1/x²) dx
= ∫(x³ + 1/x + x +1/x³) dx
= x⁴/4 + log|x| + x²/2 + x^-3+1/-3+1 + C
= x⁴/4 + log|x| + x²/2 – 1/2x² + C

Que-16: ∫(x+2)(4x²-5)/x dx

Sol: ∫(x+2)(4x²-5)/x dx
= ∫4x³+8x²-5x-10/x dx
= 4∫4x² dx + 8∫x dx – ∫5 dx – ∫10/x dx
= 4 x³/3 + 8x²/2 -5x -10log|x| + C
= 4 x³/3 + 4x² -5x -10log|x| + C

Que-17: ∫x²/4+x² dx

Sol: Let I = ∫x²/4+x² dx
= ∫4+x²-4/4+x² dx = ∫[1- 4/x²+4] dx
= x – 4 ∫dx/x²+2²
= x – 4/2 tan^-1(x/2) + C
= x – 2 tan^-1(x/2) + C

Que-18: ∫x⁴/1+x² dx

Sol: Let I = ∫x⁴/1+x² dx = ∫x⁴-1+1/1+x² dx
= ∫x⁴-1/1+x² dx + ∫dx/1+x²
= ∫(x²-1)(x²+1)/1+x² dx + ∫dx/1+x²
=x³/3 – x + tan^-1x + C

Que-19: ∫x⁶-1/1+x² dx

Sol: Let I = ∫x⁶-1/1+x² dx
=∫(x⁶+1-2)/1+x² dx
= ∫(x⁴)²+1/x²+1 dx – ∫2dx/x²+1
= ∫[(x⁴-x²+1)+ -2/x²+1] dx
= x⁵/5 – x³/3 + x -2 tan^-1x + C

Que-20: ∫(e^6logx – e^5logx)/(e^5logx – e^3logx) dx

Sol: Let I = ∫(e^6logx – e^5logx)/(e^5logx – e^3logx) dx
= ∫(e^{logx^(6)} – e^{logx^(5)})/(e^{logx^(5)} – e^{logx^(3)}) dx
= ∫x⁶ – x⁵/x⁵ – x³ dx
= ∫x⁵(x-1)/x³(x²-1) dx
= ∫x²/x+1 dx
= ∫x²-1+1/x+1 dx = ∫[(x-1)+1/x+1] dx
= x²/2 – x + log(x+1) + C

–: End of Indefinite Integrals Class 12 OP Malhotra Exe-13B ISC Math Ch-13 Solution :–

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