Class 12 RD Sharma Solutions- Chapter 20 Definite Integrals – Exercise 20.4 Part B

Evaluate of each of the following integrals (1-46):

Question 1. 

Solution:

We have,

Let

------ 1

So,,

------------ 2

Hence, by adding 1 and 2 ..

Question 2. ,

Solution:

We have,

Let, I=----- 1

So,------ 2

Adding 1 & 2 -------

Question 3. ,

Solution:

We have ,

Let,

So,

--------------- 2

Adding 1 and 2 --------

Question 4. 

Solution:

Let,------ 1

---------- 2

Adding 1 & 2 -----

Question 5. 

Solution:

Let,----------- (1)

So,

Question 6. 

Solution:

---------- 1

----- 2

Adding 1 & 2 -------

Question 7. 

Solution:

Let,

Let,

Now, x=0 ,, then

-------------------- 1

So,

------------- 2

Adding (1) and (2) ----------------

Question 8. 

Solution:

Let,

then

------------ (1)

------------------- (2)

Adding (1) & (2) ------------

Question 9.  

Solution:

Let,

Question 10. 

Solution:

Let,

Equating coefficients, we get

So,

Question 11. 

Solution:

------------- (1)

------------------ (2)

Adding (1) & (2) ----------------

Question 12. 

Solution:

Let ,-------------- 1

So,

Let,

As, x=0, t=1 ; x=π , t=-1

Hence,

Question 13. 

Solution:

Let,

Question 14. 

Solution:

we have,

=

Since, f(x) = f(-x) , f(x) is an even function.

-------------- 1

---------------- 2

Adding 1 and 2 ----------------

Now, let

Putting 2x=t, we get

Question 15. 

Solution:

Let,

Question 16. 

Solution:

We have , I=\int\limits_0^{π} \frac{x}{1+cos\alpha sinx}dx ---------- 1

------- 2

Adding 1 and 2 ----

substituting s

when x=0 , t=0 ; x=π ,

Question 17. 

Solution:

Let,

Question 18. 

Solution:

Question 19. 

Solution:

Question 20. 

Solution:

Question 21. 

Solution:

Now,

Let cosx=t

sinx dx=-dt

Question 22. 

Solution:

------------------ 1

------------ 2

Adding 1 & 2 -------------

Let ,

Question 23. 

Solution:

Let,

Here, f(x)=-f(x)

Hence, f(x) is odd function

Question 24. 

Solution:

We have,  is an even function.

Question 25. 

Solution:

we have,

Since,

this is an odd function

Question 26. 

Solution:

we have,

sin²x is even function

Hence,

Question 27. 

Solution:

Question 28. 

Solution:

we have ,

Let,

Then,

Question 29. 

Solution:

Put cosx = t then -sinx dx = dt

Question 30. 

Solution:

Let

is an odd function

Question 31. 

Solution:

Question 32. 

Solution:

Substitute π+x=u then dx=du

Question 33. 

Solution:

Let, 

Question 34. 

Solution:

Applying the property , 

Thus,

-

Question 35. 

Solution:

Let,

Question 36. 

Solution:

let tanx = v

dv = sec²xdx

Question 37. 

Solution:

Putthen

x=0 ⇒ t=0 and x=π ⇒

Question 38. 

Solution:

we know,

Also here,

f(x) = f(2π -x)

So,

Question 39. 

Solution:

then,

Question 40. If f is an integrable function such that f(2a-x)=f(x), then prove that

Solution:

We have ,

Then,

Let , 2a-t =x then dx=-dt

if t=a ⇒x=a

if t=2a ⇒ x=0

Hence Proved.

Question 41. If, prove that

Solution:

We have, 

Let 2a-t=x then dx=-dt

t=a , x=a ; t=2a , x=0

Question 42. If f is an integrable function, show that

(i)

Solution:

we have ,

clearly f(x²) is an even function .

So,

(ii)

Solution:

clearly , xf(x²) is odd function .

So,

Certainly. I'll provide a summary, 10 practice questions, and FAQs for Chapter 20: Definite Integrals, Exercise 20.4 Part B of RD Sharma's Class 12 mathematics textbook. Please note that this is based on general knowledge of the topic, as I don't have direct access to the specific textbook.

Summary:

Exercise 20.4 Part B of Chapter 20 likely focuses on more advanced techniques for evaluating definite integrals. This section probably covers:

1. Integration by parts for definite integrals

2. Definite integrals involving trigonometric functions

3. Evaluation of improper integrals

4. Application of substitution method in definite integrals

5. Definite integrals with logarithmic and exponential functions

10 Practice Questions:

1. Evaluate ∫[0 to π/2] x sin x dx using integration by parts.

2. Calculate ∫[0 to 1] x ln x dx.

3. Find the value of ∫[0 to ∞] e^(-x) dx.

4. Compute ∫[0 to π/4] sec^3 x dx.

5. Evaluate ∫[1 to e] (ln x)^2 dx.

6. Calculate ∫[0 to π/2] sin^3 x cos^2 x dx.

7. Find the value of ∫[0 to 1] x^n e^x dx, where n is a positive integer.

8. Evaluate ∫[0 to 1] √(1-x^2) dx.

9. Compute ∫[0 to π/2] sin x / (1 + cos x) dx.

10. Calculate ∫[1 to 2] (ln x) / x dx.

FAQs:

1. Q: What is integration by parts and when is it used?

A: Integration by parts is a technique used when integrating the product of two functions. It's particularly useful for integrals involving algebraic and transcendental functions.

2. Q: How do you evaluate improper integrals?

A: Improper integrals are evaluated by taking the limit of a definite integral as one or both of its limits approach infinity or a point where the integrand is undefined.

3. Q: What are some common trigonometric integrals to remember?

A: Some important ones include ∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C, and ∫ tan x dx = -ln|cos x| + C.

4. Q: How does the substitution method work for definite integrals?

A: When using substitution, you change both the integrand and the limits of integration. The new limits are found by applying the substitution to the original limits.

5. Q: What is the relationship between definite and indefinite integrals?

A: The Fundamental Theorem of Calculus connects them: if F(x) is an antiderivative of f(x), then ∫[a to b] f(x) dx = F(b) - F(a).

Would you like me to elaborate on any of these points or provide solutions to any of the practice questions?

Question 43. If f(x) is a continuous function defined on [0,2a] . Then, prove that

Solution:

We have from LHS,

substituting

we get,

Question 44. If f(a+b-x) = f(x), then prove that

Solution:

------------------[ Given that f(a+b-x) = f(x) ]

Question 45. If f(x) is a continuous function defined on [-a,a], then prove that

Solution:

we have ,

Let, x=-t, then dx=-dt

x=-a ⇒ t=a

x=0 ⇒ t=0

Hence, Proved.

Question 46. Prove that:

Solution:

Summary

Exercise 20.4 Part B of Chapter 20 likely focuses on more advanced techniques for evaluating definite integrals. This section probably covers:

• Integration by parts for definite integrals
• Definite integrals involving trigonometric functions
• Evaluation of improper integrals
• Application of substitution method in definite integrals
• Definite integrals with logarithmic and exponential functions