Integrals Class 12th Mathematics Part II CBSE Solution

 Integrals Class 12th Mathematics Part II CBSE Solution

Class 12^th Mathematics Part Ii CBSE Solution

Exercise 7.1

1. sin 2x Find an anti-derivative (or integral) of the following functions by the…
2. cos 3x Find an anti-derivative (or integral) of the following functions by the…
3. e2x Find an anti-derivative (or integral) of the following functions by the…
4. (ax + b)^2 Find an anti-derivative (or integral) of the following functions by…
5. sin 2x - 4 e3x Find an anti-derivative (or integral) of the following functions…
6. Find the following integrals. integrate (4e^3x + 1) dx
7. integrate x^2 (1- 1/x^2) dx Find the following integrals.
8. integrate (ax^2 + bx+c) dx Find the following integrals.
9. integrate 2x^2 + e^x dx Find the following integrals.
10. integrate (root x - 1/root x)^2 dx Find the following integrals.
11. integrate x^3 + 5x^2 - 4/x^2 dx Find the following integrals.
12. integrate x^3 + 3x+4/root x dx Find the following integrals.
13. integrate x^3 - x^2 + x-1/x-1dx Find the following integrals.
14. integrate (1-x) root xdx Find the following integrals.
15. integrate root x (3x^2 + 2x+3) dx Find the following integrals.
16. integrate (2x-3cosx+e^x) dx Find the following integrals.
17. integrate (2x^2 - 3sinx+5 root x) dx Find the following integrals.…
18. integrate secx (secx+tanx) dx Find the following integrals.
19. integrate sec^2x/cosec^2xdx Find the following integrals.
20. integrate 2-3sinx/cos^2xdx Find the following integrals.
21. The anti-derivative of (root x + 1/root x) equalsA. 1/3 x^1/3 + 2x^1/2 + c B.…
22. If d/dx f (x) = 4x^3 - 3/x^4 such that f(2) = 0. Then f(x) isA. x^4 + 1/x^3 -…

Exercise 7.1

Question 1.

Find an anti-derivative (or integral) of the following functions by the method of inspection.

sin 2x

Answer:

Method: To find the anti derivative of a function by inspection 
Steps: 1. In this method we look for a function whose derivative is the given function. For Example: if we need to find anti derivative of x², we know that derivative of x³ is 3x². Therefore, the variable terms comes out to be the same.
2. After that balance out the coefficients of variables by dividing and multiplying suitable terms. From above example if [we divide x³ by 3 we will get the answer as x². Hence, we can say that anti derivative of x² is x³/3.

Now, similarly,

We know that 

Therefore, the anti-derivative of sin2x is 

Question 2.

Find an anti-derivative (or integral) of the following functions by the method of inspection.

cos 3x

Answer:

We know that 

Therefore, the anti-derivative of cos3x is .

Question 3.

Find an anti-derivative (or integral) of the following functions by the method of inspection.

e^2x

Answer:

We know that 

Therefore, the anti-derivative of  is .

Question 4.

Find an anti-derivative (or integral) of the following functions by the method of inspection.

(ax + b)²

Answer:

We know that 

Therefore, the anti-derivative of  is .

Question 5.

Find an anti-derivative (or integral) of the following functions by the method of inspection.

sin 2x – 4 e^3x

Answer:

We know that

Therefore, the anti-derivative of sin2x is  …(1)

Also,

Therefore, the anti-derivative of  is  …(2)

From (1) and (2), we get,

= sin 2x – 4e^3x

Therefore, the anti-derivative of sin 2x – 4 e^3x is .

Question 6.

Find the following integrals.

Answer:

Question 7.

Find the following integrals.

Answer:

Question 8.

Find the following integrals.

Answer:

Question 9.

Find the following integrals.

Answer:

Question 10.

Find the following integrals.

Answer:

Now we know that,

∫xⁿ dx 

Therefore,

Question 11.

Find the following integrals.

Answer:

Separating the terms we get,

Applying the formula,
∫xⁿ dx = 

Answer. 
Question 12.

Find the following integrals.

Answer:

Separating the terms we get,

Applying the formula,

∫ xⁿ dx = 

Question 13.

Find the following integrals.

Answer:

Now the numerator can be factorized as,

x³ - x² + x - 1 = x²(x - 1) + 1(x - 1)

x³ - x² + x - 1 = (x² + 1)(x - 1)

Now putting this in given integral we get,

Question 14.

Find the following integrals.

Answer:

Question 15.

Find the following integrals.

Answer:

Question 16.

Find the following integrals.

Answer:

Question 17.

Find the following integrals.

Answer:

Question 18.

Find the following integrals.

Answer:

Formulas Used: ∫ sec²x dx = tanx + c and ∫ secx tanx dx = sec x + c

Opening the brackets we get,

Answer. 
Question 19.

Find the following integrals.

Answer:

= tanx –x +C

Question 20.

Find the following integrals.

Answer:

= 2tanx – 3secx + C

Question 21.

The anti-derivative of  equals
A. 

B. 

C. 

D. 

Answer:

Question 22.

If  such that f(2) = 0. Then f(x) is
A.  B. 

C.  D. 

Solution ||| The correct option is (A).

Answer:

It is given that 

Also, It is given that f(2) = 0

Therefore,