# NCERT Solutions Class 12 maths Chapter-7 (Integrals)Exercise 7.1

NCERT Solutions Class 12 Maths from class 12th Students will get the answers of Chapter-7 (Integrals)Exercise 7.1 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.

Class 12 Mathematics

Chapter-7 (Integrals)

Questions and answers given in practice

Chapter-7 (Integrals)

### Exercise 7.1

Q1. Find an anti-derivative (or integral) of the following functions by the method of inspection. sin 2x

Answer. The anti derivative of sin 2x is a function of x whose derivative is sin 2x. It is known that, $\begin{array}{l}\frac{d}{dx}\left(\mathrm{cos}2x\right)=-2\mathrm{sin}2x\\ ⇒\mathrm{sin}2x=-\frac{1}{2}\frac{d}{dx}\left(\mathrm{cos}2x\right)\\ \therefore \mathrm{sin}2x=\frac{d}{dx}\left(-\frac{1}{2}\mathrm{cos}2x\right)\end{array}$ Therefore, the anti derivative of

Q2. Find an anti derivative (or integral) of the following functions by the method of inspection. cos 3x

Answer.  The anti derivative of cos 3x is a function of x whose derivative is cos 3x. It is known that, $\begin{array}{l}\frac{d}{dx}\left(\mathrm{sin}3x\right)=3\mathrm{cos}3x\\ ⇒\mathrm{cos}3x=\frac{1}{3}\frac{d}{dx}\left(\mathrm{sin}3x\right)\\ \therefore \mathrm{cos}3x=\frac{d}{dx}\left(\frac{1}{3}\mathrm{sin}3x\right)\end{array}$ Therefore, the anti derivative of .
Q3. Find an anti derivative (or integral) of the following functions by the method of inspection. ${}^{}$e2x


Answer. The anti derivative of ${e}^{2x}$ is the function of x whose derivative is ${e}^{2x}$. It is known that, $\begin{array}{l}\frac{d}{dx}\left({e}^{2x}\right)=2{e}^{2x}\\ ⇒{e}^{2x}=\frac{1}{2}\frac{d}{dx}\left({e}^{2x}\right)\\ \therefore {e}^{2x}=\frac{d}{dx}\left(\frac{1}{2}{e}^{2x}\right)\end{array}$ Therefore, the anti derivative of
Q4. Find an anti derivative (or integral) of the following functions by the method of inspection.

Answer. The anti derivative of $\left(ax+b{\right)}^{2}$ is the function of x whose derivative is $\left(ax+b{\right)}^{2}$. It is known that $\begin{array}{l}\frac{d}{dx}\left(ax+b{\right)}^{3}=3a\left(ax+b{\right)}^{2}\\ ⇒\left(ax+b{\right)}^{2}=\frac{1}{3a}\frac{d}{dx}\left(ax+b{\right)}^{3}\\ \therefore \left(ax+b{\right)}^{2}=\frac{d}{dx}\left(\frac{1}{3a}\left(ax+b{\right)}^{3}\right)\end{array}$ Therefore, the anti derivative of

Q5. Find an anti derivative (or integral) of the following functions by the method of inspection:

Answer. The anti derivative of $\mathrm{sin}2x-4{e}^{3x}$ is the function of x whose derivative is $\mathrm{sin}2x-4{e}^{3x}$. It is known that, $\frac{d}{dx}\left(-\frac{1}{2}\mathrm{cos}2x-\frac{4}{3}{e}^{3x}\right)=\mathrm{sin}2x-4{e}^{3x}$ Therefore, the anti derivative of .

Q6. Find the following integral

Q7. Find the following integral

Q8. Find the following integral (ax2+bx+c)dx

Q9. Find the following integral (2x2+ex)dx

Q10. Find the following integral
Q11. Find the following integral

Q12. Find the following integral

Q13. Find the following integral

Q14. Find the following integral

Q15. Find the following integral

Answer. $\begin{array}{l}\int \sqrt{x}\left(3{x}^{2}+2x+3\right)dx\\ =\int \left(3{x}^{\frac{5}{2}}+2{x}^{\frac{3}{2}}+3{x}^{\frac{1}{2}}\right)dx\\ =3\int {x}^{\frac{5}{2}}dx+2\int {x}^{\frac{3}{2}}dx+3\int {x}^{\frac{1}{2}}dx\end{array}$

Q16. Find the following integral

Q17. Find the following integral (2x23sinx+5x)dx
Answer. $\int \left(2{x}^{2}-3\mathrm{sin}x+5\sqrt{x}\right)dx$
Answer. $\int \frac{{\mathrm{sec}}^{2}x}{{\mathrm{csc}}^{2}x}dx$