Class 12 Maths Chapter 7 Integrals – Exercise 7.2 NCERT Solutions

Introduction

Exercise 7.2 focuses on integration using standard results and substitution method. Students practice solving integrals of algebraic, trigonometric, exponential, and logarithmic functions by applying substitution and known formulas. This exercise is crucial for building problem‑solving skills in calculus.

Formulas Used

1. Substitution Rule: If $x=g(t)$, then

$$\int f(x)dx=\int f(g(t))g^{\mathrm{\prime }}(t)dt$$

2. Standard Integrals:

$$\int \frac{1}{1+x^2}dx={\tan }^{-1}x+C$$

$$\int \frac{1}{\sqrt{1-x^2}}dx={\sin }^{-1}x+C$$

$$\int \frac{1}{x\sqrt{x^2-1}}dx={\sec }^{-1}x+C$$

3. Exponential and Logarithmic:

$$\int e^{x}dx=e^x+C,\int \frac{1}{x}dx=\ln \mathrm{\mid }x\mathrm{\mid }+C$$

Students Frequently Make Mistakes

• Forgetting to change limits in definite integrals after substitution.

• Skipping differential term $dx=g^{\mathrm{\prime }}(t)dt$.

• Confusing inverse trigonometric integrals.

• Errors in simplification after substitution.

• Forgetting constant of integration $+C$.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. Evaluate $\int \frac{1}{1+x^2}dx$.

$${\tan }^{-1}x+C$$

Q2. Evaluate $\int \frac{1}{\sqrt{1-x^2}}dx$.

$${\sin }^{-1}x+C$$

Q3. Evaluate $\int \frac{1}{x\sqrt{x^2-1}}dx$.

$${\sec }^{-1}x+C$$

Q4. Evaluate $\int e^{x}dx$.

$$e^x+C$$

Q5. Evaluate $\int \frac{1}{x}dx$.

$$\ln \mathrm{\mid }x\mathrm{\mid }+C$$

Q6. Evaluate $\int \cos xdx$.

$$\sin x+C$$

Q7. Evaluate $\int \sin xdx$.

$$-\cos x+C$$

Q8. Evaluate $\int \frac{1}{a^2+x^2}dx$.

$$\frac{1}{a}{\tan }^{-1}\left(\frac{x}{a}\right)+C$$

Q9. Evaluate $\int \frac{1}{\sqrt{a^2-x^2}}dx$.

$${\sin }^{-1}\left(\frac{x}{a}\right)+C$$

Q10. Evaluate $\int \frac{1}{x\sqrt{x^2-a^2}}dx$.

$$\frac{1}{a}{\sec }^{-1}\left(\frac{x}{a}\right)+C$$

FAQs (10)

FAQ1. What is substitution method? Replacing variable with another to simplify integral.

FAQ2. Why change limits in definite integrals? Because variable substitution changes domain.

FAQ3. What is integral of $\frac{1}{1+x^2}$? ${\tan }^{-1}x+C$.

FAQ4. What is integral of $\frac{1}{\sqrt{1-x^2}}$? ${\sin }^{-1}x+C$.

FAQ5. What is integral of $\frac{1}{x\sqrt{x^2-1}}$? ${\sec }^{-1}x+C$.

FAQ6. What is integral of $e^x$? $e^x+C$.

FAQ7. What is integral of $\frac{1}{x}$? $\ln \mathrm{\mid }x\mathrm{\mid }+C$.

FAQ8. Why add constant $C$? Indefinite integrals represent family of functions.

FAQ9. What is integral of $\cos x$? $\sin x+C$.

FAQ10. Why is Exercise 7.2 important? It builds foundation for substitution and inverse trigonometric integrals.

Conclusion

Exercise 7.2 has 10 solved questions and 10 FAQs that strengthen your understanding of substitution and standard integrals. This builds the foundation for advanced integration techniques in Class 12 Maths.

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