Class 12 NCERT Solutions- Mathematics Part ii – Chapter 7– Integrals Exercise 7.7

In the article, we will solve Exercise 7.7 from Chapter 7, “Integrals” in the NCERT. Exercise 7.7 covers some special types of standard integrals which are based on integration by parts

Integral Formulae To Solve Exercise 7.7

...(i)

...(iI)

...(iII)

Exercise 7.7 Solutions

Q.1:

Solution:

by putting a = 2 in formula (1), we get

Q.2:

Solution:

by putting

now by using formula (1), we get

now we put y = 2x

Q.3:

Solution:

let (x+2) = y and dx = dy

using formula (2) , we get

now putting y = (x+2)

Q.4:

Solution:

substitute (x+2) = y and dx = dy

Q.5:

Solution:

now putting (x+2) = y and dx = dy

using formula (1)

now putting (x+2) =y

Q.6:

Solution:

now substitute (x+2) = y and dx = dy

now put (x+2) = y

Q.7:

Solution:

now substitute , and dx = dy

using formula (1), we get

Q.8:

Solution:

now substitute and dx = dy

using formula (3) , we get

Q.9:

Solution:

by using formula (2)

Q.10:

(A) x/2√(1 + x²) + 1/2log|x + √(1 + x²) + C|

(B) 2/3(1 + x²)^3/2 + C

(C) 2/3x√(1 + x²)^3/2 + C

(D) x²/2√(1 + x²) + 1/2.x².log|x + √(1 + x²) + C|

Solution:

Option (A) is Correct

Using formula (2) we get

Q.11:

(A) 1/2(x - 4)√(x² - 8x + 7) + 9.log|x - 4 + √(x² -8x + 7)| + C

(B) 1/2(x - 4)√(x² - 8x + 7) + 9.log|x + 4 + √(x² -8x + 7)| + C

(C) 1/2(x - 4)√(x² - 8x + 7) - 3√(2).log|x - 4 + √(x² -8x + 7)| + C

(D) 1/2(x - 4)√(x² - 8x + 7) - 9/2.log|x - 4 + √(x² -8x + 7)| + C

Solution:

Option (D) is Correct

let (x-4) = y and dx = dy

using formula (3)

Now put y = (x-4) and after simplifying