Integrals

page -299

Question 1:sin 2x

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Answer:

The anti derivative of sin 2x is a function of x whose derivative is sin 2x.

It is known that,

Therefore, the anti derivative of

Question 2:

Cos 3x

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Answer:

The anti derivative of cos 3x is a function of x whose derivative is cos 3x.

It is known that,

Therefore, the anti derivative of .

Question 2:

Cos 3x

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Answer:

The anti derivative of cos 3x is a function of x whose derivative is cos 3x.

It is known that,

Therefore, the anti derivative of .

Question 3:

e^2x

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Answer:

The anti derivative of e^2x is the function of x whose derivative is e^2x.

It is known that,

Therefore, the anti derivative of .

Question 4:

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Answer:

The anti derivative of is the function of x whose derivative is .

It is known that,

Therefore, the anti derivative of .

Question 5:

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Answer:

The anti derivative of  is the function of x whose derivative is .

It is known that,

Therefore, the anti derivative of  is .

Question 6:

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Answer:

Question 7:

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Answer:

Question 8:

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Answer:

Question 9:

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Answer:

Question 10:

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Answer:

Question 11:

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Answer:

Question 12:

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Answer:

Question 13:

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Answer:

On dividing, we obtain

Question 14:

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Answer:

Question 15:

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Answer:

Question 16:

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Answer:

Question 17:

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Answer:

Question 18:

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Answer:

Question 19:

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Answer:

Question 20:

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Answer:

Question 21:

The anti derivative of equals

(A) (B) 

(C) (D)

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Answer:

Hence, the correct answer is C.

Question 22:

If such that f(2) = 0, then f(x) is

(A) (B) 

(C) (D)

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Answer:

It is given that,

∴Anti derivative of 

∴

Also,

Hence, the correct answer is A.

Page-304

Question 1:

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Answer:

Let = t

∴2x dx = dt

Question 2:

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Answer:

Let log |x| = t

∴ 

Question 3:

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Answer:

Let 1 + log x = t

∴ 

Question 4:

sin x ⋅ sin (cos x)

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Answer:

sin x ⋅ sin (cos x)

Let cos x = t

∴ −sin x dx = dt

Question 5:

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Answer:

Let 

∴ 2adx = dt

Question 6:

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Answer:

Let ax + b = t

⇒ adx = dt

Question 7:

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Answer:

Let 

∴ dx = dt

Question 8:

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Answer:

Let 1 + 2x² = t

∴ 4xdx = dt

Question 9:

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Answer:

Let 

∴ (2x + 1)dx = dt

Question 10:

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Answer:

Let 

∴

Question 11:

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Answer:

Let 

∴ dx = dt

Question 12:

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Answer:

Let 

∴ 

Question 13:

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Answer:

Let 

∴ 9x² dx = dt

Question 14:

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Answer:

Let log x = t

∴ 

Question 15:

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Answer:

Let 

∴ −8x dx = dt

Question 16:

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Answer:

Let 

∴ 2dx = dt

Question 17:

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Answer:

Let 

∴ 2xdx = dt

Pg 305

Question 18:

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Answer:

Let 

∴ 

Question 19:

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Answer:

Dividing numerator and denominator by e^x, we obtain

Let 

∴ 

Question 20:

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Answer:

Let 

∴ 

Question 21:

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Answer:

Let 2x − 3 = t

∴ 2dx = dt

Question 22:

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Answer:

Let 7 − 4x = t

∴ −4dx = dt

Question 23:

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Answer:

Let 

∴ 

Question 24:

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Answer:

Let 

∴ 

Question 25:

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Answer:

Let 

∴ 

Question 26:

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Answer:

Let 

∴ 

Question 27:

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Answer:

Let sin 2x = t

∴ 

Question 28:

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Answer:

Let 

∴ cos x dx = dt

Question 29:

cot x log sin x

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Answer:

Let log sin x = t

Question 30:

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Answer:

Let 1 + cos x = t

∴ −sin x dx = dt

Question 31:

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Answer:

Let 1 + cos x = t

∴ −sin x dx = dt

Question 32:

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Answer:

Let sin x + cos x = t ⇒ (cos x − sin x) dx = dt

Question 33:

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Answer:

Put cos x − sin x = t ⇒ (−sin x − cos x) dx = dt

Question 34:

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Answer:

Question 35:

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Answer:

Let 1 + log x = t

∴ 

Question 36:

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Answer:

Let 

∴ 

Question 37:

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Answer:

Let x⁴ = t

∴ 4x³ dx = dt

Let 

∴

From (1), we obtain

Question 38:

equals

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Answer:

Let 

∴ 

Hence, the correct answer is D.

Question 39:

equals

A.

B.

C.

D.

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Answer:

Hence, the correct answer is B.