## EXERCISE 1.12

#### Page No 352:

#### Question 1:

#### Answer:

Equating the coefficients of *x*^{2}, *x*, and constant term, we obtain

−*A* +* B *− *C* = 0

*B* + *C *= 0

*A* = 1

On solving these equations, we obtain

From equation (1), we obtain

#### Question 2:

#### Answer:

#### Question 3:

[Hint: Put]

#### Answer:

#### Question 4:

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#### Question 5:

#### Answer:

On dividing, we obtain

#### Question 6:

#### Answer:

Equating the coefficients of *x*^{2}, *x*, and constant term, we obtain

*A* + *B* = 0

*B *+ *C* = 5

9*A* + *C *= 0

On solving these equations, we obtain

From equation (1), we obtain

#### Question 7:

#### Answer:

Let *x *−* a *= *t *⇒ *dx* = *dt*

#### Question 8:

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#### Question 9:

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Let sin *x* = *t* ⇒ cos *x dx* = *dt*

#### Question 10:

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#### Question 11:

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#### Question 12:

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Let *x*^{4 }=* t* ⇒ 4*x*^{3} *dx* = *dt*

#### Question 13:

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Let *e*^{x} = *t* ⇒ *e*^{x} *dx* = *dt*

#### Question 14:

#### Answer:

Equating the coefficients of *x*^{3}, *x*^{2}, *x*, and constant term, we obtain

*A* + *C* = 0

*B* + *D* = 0

4*A* + *C* = 0

4*B *+ *D* = 1

On solving these equations, we obtain

From equation (1), we obtain

#### Question 15:

#### Answer:

= cos^{3} *x* × sin *x*

Let cos *x* =* t* ⇒ −sin *x dx* =* dt*

#### Question 16:

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#### Question 17:

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#### Question 18:

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#### Question 19:

#### Answer:

Let I=∫sin-1x-cos-1xsin-1x+cos-1xdx

It is known that, sin-1x+cos-1x=π2

⇒I=∫π2-cos-1x-cos-1xπ2dx

=2π∫π2-2cos-1xdx

=2π.π2∫1.dx-4π∫cos-1xdx

=x-4π∫cos-1xdx …(1)

Let I1=∫cos-1x dx

Also, let x=t⇒dx=2 t dt

⇒I1=2∫cos-1t.t dt

=2cos-1t.t22-∫-11-t2.t22dt

=t2cos-1t+∫t21-t2dt

=t2cos-1t-∫1-t2-11-t2dt

=t2cos-1t-∫1-t2dt+∫11-t2dt

=t2cos-1t-t21-t2-12sin-1t+sin-1t

=t2cos-1t-t21-t2+12sin-1t

From equation (1), we obtain

I=x-4πt2cos-1t-t21-t2+12sin-1t =x-4πxcos-1x-x21-x+12sin-1x

=x-4πxπ2-sin-1x-x-x22+12sin-1x

#### Question 20:

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#### Question 21:

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#### Question 22:

#### Answer:

Equating the coefficients of *x*^{2}, *x*,and constant term, we obtain

*A* + *C* = 1

3*A* + *B* + 2*C *= 1

2*A* + 2*B* + *C* = 1

On solving these equations, we obtain

*A* = −2, *B* = 1, and *C* = 3

From equation (1), we obtain

#### Page No 353:

#### Question 23:

#### Answer:

#### Question 24:

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Integrating by parts, we obtain

#### Question 25:

#### Answer:

#### Question 26:

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When *x *= 0, *t *= 0 and

#### Question 27:

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When and when

#### Question 28:

#### Answer:

When and when

As , therefore, is an even function.

It is known that if *f*(*x*) is an even function, then

#### Question 29:

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#### Question 30:

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#### Question 31:

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From equation (1), we obtain

#### Question 32:

#### Answer:

Adding (1) and (2), we obtain

#### Question 33:

#### Answer:

From equations (1), (2), (3), and (4), we obtain

#### Question 34:

#### Answer:

Equating the coefficients of *x*^{2}, *x*, and constant term, we obtain

*A* + *C* = 0

*A* + *B* = 0

*B* = 1

On solving these equations, we obtain

*A* = −1, *C* = 1, and *B* = 1

Hence, the given result is proved.

#### Question 35:

#### Answer:

Integrating by parts, we obtain

Hence, the given result is proved.

#### Question 36:

#### Answer:

Therefore, *f* (*x*) is an odd function.

It is known that if *f*(*x*) is an odd function, then

Hence, the given result is proved.

#### Question 37:

#### Answer:

Hence, the given result is proved.

#### Question 38:

#### Answer:

Hence, the given result is proved.

#### Question 39:

#### Answer:

Integrating by parts, we obtain

Let 1 − *x*^{2} = *t* ⇒ −2*x* *dx* = *dt*

Hence, the given result is proved.

#### Question 40:

Evaluate as a limit of a sum.

#### Answer:

It is known that,

#### Question 41:

is equal to

**A.**

**B.**

**C.**

**D. **

#### Answer:

Hence, the correct answer is A.

#### Question 42:

is equal to

**A.**

**B.**

**C.**

**D. **

#### Answer:

Hence, the correct answer is B.

#### Page No 354:

#### Question 43:

If then is equal to

**A.**

**B.**

**C.**

**D. **

#### Answer:

Hence, the correct answer is D.

#### Question 44:

The value of is

**A.** 1

**B.** 0

**C.** − 1

**D. **

#### Answer:

Adding (1) and (2), we obtain

Hence, the correct answer is B.