CBSE Guess > Papers > Important Questions > Class XII > 2010 > Maths > Mathematics By Mr. M.P.Keshari

Integrals

Q.4. Evaluate : ∫tan ^-1√{(1 – sin x)/(1 + sin x)}dx.

Solution :

We have, ∫tan ^-1 √{(1 – sin x)/(1 + sin x)}dx = I (say)
Now, (1 – sin x)/(1 + sin x)
= (cos2 x/2 + sin² x/2 – 2sin x/2.cos x/2)/(cos² x/2 + sin² x/2 + 2sin x/2.cos x/2)
= (cos x/2 – sin x/2)/(cos x/2 + cos x/2)
= (1 – tan x/2)/(1 + tan x/2)
= tan (π/4 – x/2).
Therefore, I = ∫tan –1 [tan (π/4 – x/2)] dx
= ∫(π/4 – x/2) dx = π/4∫dx – ∫xdx/2
= π/4 x – x2/4 + c. [Ans.]

Q.5. Evaluate : ∫sin x √(1 + cos²x) dx.

Solution :

We have I = ∫ sin x √(1 + cos²x) dx
= ∫ sin x √(2cos² x) dx
= √2∫si x cos x dx
Put sin x = t => cos x dx = dt
Therefore, I = √2∫t dt = (√2)t²/2 + c = 1/√2 sin² x + c . [Ans.]
Or, I = √2∫sin x cos x dx
= 1/√2∫sin²x dx
= 1/√2( – cos 2x)/²
= – 1/2√2 cos²x + c. [Ans.]

Q.6. Evaluate : ∫[cos (x + a)/sin (x + b)] dx.

Solution :

Let I = ∫[cos (x + a)/sin (x + b)] dx
= ∫[cos {(x + b) +(a – b)}/{sin(x + b)}] dx
= ∫[{cos (x + b) cos (a – b) – sin (x +b) sin (a – b)}/{sin (x + b)}] dx
= cos (a – b)∫cot (x + b) dx – sin (a – b)∫1. dx
= cos (a – b) log|sin (x + b)| – x sin (a – b) + c. [Ans.]

Q.7. Evaluate : ∫[(2cos x)/(3sin2 x)] dx.

Solution :

Let I = ∫[2cos x/3sin2 x] dx
= 2/3∫[(1/sin x)(cos x/sin x) dx
= 2/3∫[cosec x cot x] dx
= – 2/3 cosec x + c. [Ans.]

Paper By Mr. M.P.Keshari
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