# NCERT Solutions Class 12 maths Chapter-7 (Integrals)Exercise 7.2

NCERT Solutions Class 12 Maths from class 12th Students will get the answers of Chapter-7 (Integrals)Exercise 7.2 This chapter will help you to learn the basics and you should expect at least one question in your exam from this chapter.
We have given the answers of all the questions of NCERT Board Mathematics Textbook in very easy language, which will be very easy for the students to understand and remember so that you can pass with good marks in your examination.

### Exercise 7.2

Q1. Integrate the function $\frac{2x}{1+{x}^{2}}$.

Q2. Integrate the function

Answer.  Let log |x| = t , $\therefore \frac{1}{x}dx=dt$

Q3. Integrate the function

Answer. $\begin{array}{}\end{array}$1x+xlogx=1x(1+logx) Let 1+logx=t1xdx=dt

Q4. Integrate the function: sin x ⋅ sin (cos x)

Answer. sin x ⋅ sin (cos x) Let cos x = t ∴ −sin x dx = dt
Q5. Integrate the function sin (ax + b ) cos ( ax + b )

Answer. $\mathrm{sin}\left(ax+b\right)\mathrm{cos}\left(ax+b\right)=\frac{2\mathrm{sin}\left(ax+b\right)\mathrm{cos}\left(ax+b\right)}{2}=\frac{\mathrm{sin}2\left(ax+b\right)}{2}$

Q6. Integrate the function

$\begin{array}{}\end{array}$ Let ax+b=tadx=dtdx=1adt
Q7. Integrate the function xx+2

Answer. Let (x + 2) = t ∴ dx = dt $\begin{array}{rl}⇒\int x\sqrt{x+2}dx& =\int \left(t-2\right)\sqrt{t}dt\\ & =\int {t}^{\frac{3}{2}}-2{t}^{\frac{1}{2}}\right)dt\\ & =\int {t}^{\frac{3}{2}}dt-2\int {t}^{\frac{1}{2}}dt\end{array}$

Q8. Integrate the function
Answer. $\begin{array}{}\end{array}$ Let 1+2x2=t4xdx=dt
Q9. Integrate the function: (4x+2)x2+x+1


Answer.  $\begin{array}{rl}& \int \left(4x+2\right)\sqrt{{x}^{2}+x+1}dx\\ & =\int 2\sqrt{t}dt\\ & =2\int \sqrt{t}dt\end{array}$
Q10. Integrate the function :

Q11. Integrate the function :

Answer. Let x + 4 = t ∴ dx = dt $\begin{array}{rl}\int \frac{x}{\sqrt{x+4}}dx& =\int \frac{\left(t-4\right)}{\sqrt{t}}dt\\ & =\int \left(\sqrt{t}-\frac{4}{\sqrt{t}}\right)dt\\ & =\frac{{t}^{\frac{3}{2}}}{\frac{3}{2}}-4\left(\frac{\frac{1}{2}}{\frac{1}{2}}\right)+\mathrm{C}\end{array}$
Q12. Integrate the function: (x31)13x5


Answer.  $\begin{array}{rl}⇒\int {\left({x}^{3}-1\right)}^{\frac{1}{3}}{x}^{5}dx& =\int {\left({x}^{3}-1\right)}^{\frac{1}{3}}{x}^{3}\cdot {x}^{2}dx\\ & =\int {t}^{\frac{1}{3}}\left(t+1\right)\frac{dt}{3}\\ & =\frac{1}{3}\int {t}^{\frac{4}{3}}+{t}^{\frac{1}{3}}\right)dt\end{array}$

Q13. Integrate the function :

Q14. Integrate the function : 1x(logx)m,x>0,m1