Indefinite Integrals Class 12 OP Malhotra Exe-13A ISC Maths Solutions Ch-13 Solutions. In this article you would learn about constant of integration and two general theorems on indefinite integrals . 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-13A ISC Maths Solutions Ch-13
| Board | ISC |
| Publications | S Chand |
| Subject | Maths |
| Class | 12th |
| Chapter-13 | Indefinite Integrals |
| Writer | OP Malhotra |
| Exe-13(a) | constant of integration and two general theorems on indefinite integrals |
Exercise- 13A
Indefinite Integrals Class 12 OP Malhotra Exe-13A Solution.
Que-1: (i) sin 2x
(ii) 2 sin 3x
(iii) 1/3cos4x
(iv) cos 5x / 2
(v) 8 cos2 8
(vi) cosec2 2x
(vii) sec 5x tan 5x
(viii) -cosec 3x cot 3x
Sol:
Que-2: (i) cos (5 – 3x)
(ii) 2sin (π/2-x²)
(iii) sin(3/4 x + 5)
(iv) 4 sec2(2x – 4)
Sol:
Que-3: (i) sin2 x
(ii) cos2 x
(iii) sin3 x
(iv) sin2 mx
(v) sin2 x cos2 x
(vi) sin3 x cos3 x
(vii) cos 2x + 2x sin²x / cos²x
(viii) sin x sec2 x
(ix) sin3 x cos3 x
(x) 1/sin²x cos²x
(xi) sec x/secx + tanx
(xii) 3 cosec2 x + 2 sin3x
Sol:
Que-4: (i) cos 4x cos 3x dx
(ii) sin 4xsin 8x
Sol:
Que-5: (i) cos2x . cos4x . cos6x
(ii) sin x . sin2x . sin3x
(iii) cos²x-sin²x/√1+cos4x
(iv) cos4 xsin4x
(v) 7 cos³x + 8 sin³x / 3sin²x cos²x
Sol:
Que-6: (i) 1/1+cosx
(ii) 1/1-cos2x
(iii) 1/1-sinx
(iv) 1-cos2x/1+cos2x
(v) √1+cosx
(vi) √1+sin2x
(vii) cosx√1+cos2x
(viii) sinx√1-cos2x
(ix) cosx-sinx/cosx+sinx (2+2sin2x)
(x) 4-5sinx/cos²x + 1/sin²x.cos²x
(xi) sinx+cosx/√1+sin2x
Sol:
Que-7: √(1+sin x/2)
Sol:
Que-8: sin6x + sin6x / sin²x . cos²x
Sol:
Que-9: sin6x
Sol:
Que-10: tan-1(sin2x/1+cos2x)
Sol:
Que-11: cos-1(1-tan²x/1+tan²x)
Sol:
Que-12: cos-1(sin x)
Sol:
Que-13: If f ‘ (x) = 3x2 – 2/x³ and f (1) = 0, find f(1).
Sol: Given f ‘ (x) = 3x2 – 2/x³;
on integrating both sides, we have
f(x) = 3x³/3 – 2 x-3+1/-3+1 + C
⇒ f(x) = x3 + 1/x² + C …(1)
since f(1) = 0 i.e. when x = 1, f(x) = 0
∴ from (1) ; 0 = 1 + 1/1 + C
⇒ C = -2
∴ eqn (1) gives ; f(x) = x3 + 1/x² – 2
Que-14: If f ‘ (x) = a sin x + b cos x and f ‘ (0) = 4, f(0) = 3, f(π/2) = 5, find f(x)
Sol: Gives f ‘ (x) = a sin x + b cos x
Since f ‘ (0) = 4
∴ from (1) ; 4 = a × 0 + b × 1
⇒ b = 4
Also f(x) = ∫f'(x)dx+C
⇒ f(x) = ∫(asinx+bcosx)dx+C
⇒ f(x) = – a cos x + b sin x + C …(2)
since f(0) = 3 i.e. when x = 0; f(x) = 3
∴ from (2); 3 = – a \times 1 + b × 0 + C
⇒ 3 = – a + C
Also f(π/2) = 5 i.e. when x = (π/2),(x) = 5
∴ from (2); 5 = – a × 0 + b × 1 + C
⇒ 5 = b + C ⇒ 5 = 4 + C
⇒ C = 1
∴ from (3); 3 = – a + 1 ⇒ a = – 2
∴ from (2); we have
f(x) = – 2 cos x + 4 sin x + 1
–: End Indefinite Integrals Class 12 OP Malhotra Exe-13A ISC Math Ch-13 Solution :–
Return to :- OP Malhotra ISC Class-12 S Chand Publication Maths Solutions
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