Important Questions for Class 12 Chapter 7 - Integrals

We have come up with CBSE important questions for class 12th chapter 7 Integrals with a lot of practice questions. A class 12th student clearly understands the importance of practicing any concept learned. Further, these list of questions are available for free. Download now!

Important questions based on NCERT syllabus for Class 12 Chapter 7 - Integrals:

Question 1: Integrate (8x+7) e^x
Solution: ∫(8x+7) e^x

Thsi equation of the form ∫u dv = u.v - ∫v du

Here u = 8x+ 7 dv = e^x
du = 8 v = ∫e^x
= e^x
∫u dv = u.v - ∫v du
∫(8x+7) e^x = e^x(8x+7) - ∫8e^x
= e^x(8x+7) - 8e^x + C where C is the integration constant.

Question 2: integrate ∫(1/xlogx) dx
Solution: Consider ∫(1/xlogx) dx
we add and subtract log in the numarator we get
∫(1 + log x - log x)/xlogx = ∫(1+log x)/xlog x - ∫logx/xlogx
= ∫(1+log x)/xlog x - ∫1/x dx
= ∫(1+log x)/xlog x - log|x| + C
Now we have to integrate (1+log x)/xlog x, for that we substitute
u = xlogx then du becomes x.1/x + logx(1) = 1 + logx
hence ∫(1+log x)/xlog x = ∫ du/u
= log|u| + C
= log|xlogx| + C
= log|x| + log|logx| + C

Therefore (1 + log x - log x)/xlogx = log|x| + log|logx| - log|x| + C
= log|logx| + C

Question 3: Integrate (e^(2x)-e^(-2x))/(e^(2x) + e^(-2x))
Solution: Let I = (e^(2x)-e^(-2x))/(e^(2x) + e^(-2x)).....(1)

Put e^(2x) + e^(-2x) = t
dt = 2e^(2x) - 2e^(-2x)
= 2(e^(2x) - e^(-2x))
dt/2 = e^(2x) - e^(-2x)

Substitute in equation 1 we get

I = ∫ (1/t) (dt/2)
= 1/2 ∫ dt/t
= 1/2(log|t|) + C
= 1/2(log(e^(2x) + e^(-2x)) + C