ROUTERA


Chapter 7 Integrals

Class 12th Maths NCERT Exemplar Solution



Exercise
Question 1.

Verify the following :



Answer:

To Verify;




Let t = 2x + 3








Hence Verified



Question 2.

Verify the following :



Answer:

To Verify;


Let; t=x2 + 3x


⇒ dt = 2x + 3





[∵t = x2 + 3x]


Hence Verified



Question 3.

Evaluate the following:



Answer:

Given;

Let t = x + 1


⇒ dx = dt










Question 4.

Evaluate the following:



Question has been changed


Answer:

Given;

As we know n log x = log xn




Take x3 common out of numerator and denominator to get,





= ∫ x2 dx



So,




Question 5.

Evaluate the following:



Answer:

Given;

Let t = x + sin x


⇒ dt = (1 + cos x) dx



=log |t|+C


=log |x + sin x |+C



Question 6.

Evaluate the following:



Answer:

Given;






As we know,


∫cosec x cot x dx=−cosec x+c


∫cosec2x dx=−cot x+c




Question 7.

Evaluate the following:



Answer:

Given; ∫tan2 x sec4 x dx

=∫ tan2 x sec2 x (1+ tan2 x) dx


Let; tan x = y


⇒ sec2 x dx = dy


=∫y2+y4 dy





Question 8.

Evaluate the following:



Answer:

Given;




= x + C



Question 9.

Evaluate the following:



Answer:

Given;







Question 10.

Evaluate the following:

(Hint: Put √x = z)


Answer:

Given;

Let z = √x


⇒ x = z2


⇒ dx = 2z dz




Let; t = z + 1


i.e. t = √x + 1


⇒ dt = dz









Question 11.

Evaluate the following:



Answer:

Given;




Let t=a2 -x2


⇒-2x dx=dt



and





Question 12.

Evaluate the following:

(Hint : Put x = z4)


Answer:

Given;

Let x = z4


⇒ dx = 4z3 dz





Let t = 1 + z3 [i.e. t = 1 + x3/4]


⇒ dt = 3z2 dz







Question 13.

Evaluate the following:



Answer:

Given;

Let x = tan y


⇒ dx = sec2 y dy





Let t = sin y


⇒ dt = cos y dy








Question 14.

Evaluate the following:



Answer:

Given;





Question 15.

Evaluate the following:



Answer:

Given;







Question 16.

Evaluate the following:



Answer:

Given;


[Let; x2 + 9 = y ⇒ 2x dx = dy]





Question 17.

Evaluate the following:



Answer:

Given;





Question 18.

Evaluate the following:



Answer:

Given;

[Let; t = x2⇒ dt = 2x dx]











Question 19.

Evaluate the following:



Answer:

Given;






As we know,







Question 20.

Evaluate the following:



Answer:

Given;





Question 21.

Evaluate the following:



Answer:

Given;






Apply integration by parts









Question 22.

Evaluate the following:



Answer:

Given;











Question 23.

Evaluate the following:



Answer:

Given;








Question 24.

Evaluate the following:



Answer:

Given;






Question 25.

Evaluate the following:



Answer:

Given;







Question 26.

Evaluate the following:



Answer:

Given;






Question 27.

Evaluate the following as limit of sums:



Answer:

Given;

We know


Here a = 0 , b = 2



⇒ nh = 2












Question 28.

Evaluate the following as limit of sums:



Answer:

We know


Here a = 0 , b = 2



⇒ nh = 2








= e2 - 1



Question 29.

Evaluate the following:



Answer:

Given;







Question 30.

Evaluate the following:



Answer:

Given;





By applying partial fraction;




When u = −1;













By applying the given limits 0 to π/2





Question 31.

Evaluate the following:



Answer:

Given



Using perfect square method for the denominator





We know that



=sin-1(1)-sin-1 (-1)


We know sin-1 (-θ ) = - sin θ



= π



Question 32.

Evaluate the following:



Answer:

Given

Now put


⇒ 2x dx=dt


At x=0, t=1 and


at x=1, t=2







Question 33.

Evaluate the following:



Answer:

Using Property


Let … (1)



⇒As sin(π-x) = sin x and


cos(π-x) = - cos x






Now let cos x=t


⇒- sin x dx=dt


And, at x=0, t=1


and at x=π, t=-1







Question 34.

Evaluate the following:

(Hint: let x = sin θ)


Answer:

Given


⇒Let



At x=0, θ=0









⇒As sec2 θ-tan2 θ=1



Now put tan θ=t


⇒sec2 θ dθ=dt


At θ=0, t=0




As






Question 35.

Evaluate the following:



Answer:

Given:


Put




(Concept of partial fraction)



On comparing coefficients of ‘t’ we get




⇒ Now put back in the above eq.




⇒ Now





Question 36.

Evaluate the following:



Answer:

Given


Put



(Concept of partial fraction)



On comparing coefficients of ‘t’ we get




⇒ Now put back in the above eq.




⇒ Now






Question 37.

Evaluate the following:



Answer:

Given


Let


Now using Property










Put



⇒ At






Question 38.

Evaluate the following:



Answer:

Given:


Using the concept of partial fractions,




Comparing coefficients:












Question 39.

Evaluate the following:



Answer:

Given:


Put




⇒ As




Now using the property:








Question 40.

Evaluate the following:

(Hint: Put x = a tan2 θ)


Answer:

Given:





⇒ As




















Question 41.

Evaluate the following:



Answer:

Given:


Using Trigonometric identities:


⇒ cos2x=2 cos2x-1= 1-2 sin2 x
















Question 42.

Evaluate the following:



Answer:

Given:


Using trigonometric identity














Question 43.

Evaluate the following:

(Hint: Put tan x = t2)


Answer:

Given


Put







Taking out common in both the numerators




⇒ Now










Put (2) and (3) in (1)







Question 44.

Evaluate the following:



(Hint: Divide Numerator and Denominator by cos4x)


Answer:

Given:


Dividing Numerator and Denominator by cos4x





⇒ Put






















Question 45.

Evaluate the following:



Answer:

Given


Let






⇒ Apply Integration by parts




⇒ Put (2) and (3) in (1)





Question 46.

Evaluate the following:



Answer:

Given:


Using the property:


Let


-





Using the property:



⇒ Let ..(3)


⇒ Using the property:








⇒ 2dx=dt and limits changes from 0to π



⇒ from equation (2)again becomes,



⇒ From eq. (3)



..(7)


⇒ On putting (7) in (2) and the obtained result in(1)





Question 47.

Evaluate the following:



Answer:

Given:


Let


Using the property:





Adding equation (1) and (2)




⇒ Put 2x=t


⇒ 2xdx=dt



⇒ As cos(-x) = cos x


Using property:



Using the property:



⇒ Now From previous question eq.(7) we obtained





Question 48.

is equal to
A. 2(sinx + xcos θ) + C

B. 2(sinx – xcos θ) + C

C. 2(sinx + 2xcos θ) + C

D. 2(sinx – 2x cos θ) + C


Answer:

Using Trigonometric identity





=2∫(cos x +cos θ)dx


=2∫cosxdx+2∫cosθdx



Question 49.

is equal to
A.

B.

C.

D.


Answer:

Given:


Multiply Nr and Dr by




⇒ Also










Question 50.

is equal to
A.

B.

C.

D.


Answer:

Given:


Put





⇒ Now apply integration by part on ∫t tan-1 t dt






⇒ Put (3) in (2) and the resulting equation in (1)







Question 51.

is equal to
A.

B.

C.

D.


Answer:

Given:





Now using the property:







Question 52.

is equal to
A.

B.

C.

D.


Answer:

Given:


Taking x2 out from the denominator




⇒ Now put





Question 53.

If log |x + 2| + C, then
A.

B.

C.

D.


Answer:

Given: …(1)


Using concept of partial fractions





⇒ A+B=0 …(1)


⇒ C+2B=0 …(2)


⇒ A+2C=1 …(3)


⇒ On solving the above three equations we get








On comparing (1) and (2) we get,



Question 54.

is equal to
A.

B.

C.

D.


Answer:

Given:






Question 55.

is equal to
A. log |1 + cos x| + C

B. log |x + sin x| + C

C.

D.


Answer:

D.

Given:


As we know








⇒ ∫(2t+2t tan2 t+2 tan t) dt=2∫(t + t tan^2 t+ tan t) dt


⇒ 2∫tdt+2∫t (sec2(t-1)dt+2∫tan t dt


⇒ 2∫tdt+2∫t sec2 t dt-2∫tdt+2∫tan t dt


⇒ 2∫t sec2 t dt+2∫tant dt ….(1)


Applying Integration by parts on ∫t sec2 t dt



⇒ ∫t sec2 t dt=t tan t-∫tan t dt ….(2)


⇒ Put (2) in(1)




Question 56.

If , then
A.

B.

C.

D.


Answer:

Given: …(1)


Put









…(3)


Comparing (1) and (3)



Question 57.

is equal to
A. 1

B. 2

C. 3

D. 4


Answer:

Given:


Using trigonometric identities:








Question 58.

is equal to
A. 2√2

B. 2(√2 + 1)

C. 2

D. 2 (√2 –1)


Answer:

As







⇒ On solving the Above Integral we get


Question 59.

Fill in the blanks in each of the following

is equal to ___________.


Answer:

e-1

Given:


Put



⇒ At




Question 60.

Fill in the blanks in each of the following

___________.


Answer:


Given





Now using the property:








Question 61.

Fill in the blanks in each of the following

If then a = ____________.


Answer:


Given:












Question 62.

Fill in the blanks in each of the following

__________.


Answer:


Given










Question 63.

Fill in the blanks in each of the following

The value of is ____________.


Answer:

0

Using the property:


Let