Class 12 Maths Chapter 7 Integrals – Exercise 7.1 NCERT Solutions

Introduction

Exercise 7.1 introduces the concept of indefinite integrals as the reverse process of differentiation. Students learn basic integration formulas, rules of integration, and how to apply them to algebraic and trigonometric functions. This exercise builds the foundation for definite integrals and applications.

Formulas Used

1. Basic Integration Formulas:

$$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C,n\mathrm{\ne }-1$$

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

$$\int e^{x}dx=e^x+C$$

$$\int \sin xdx=-\cos x+C$$

$$\int \cos xdx=\sin x+C$$

$$\int {\sec }^{2}xdx=\tan x+C$$

$$\int {\csc }^{2}xdx=-\cot x+C$$

$$\int \sec x\tan xdx=\sec x+C$$

$$\int \csc x\cot xdx=-\csc x+C$$

2. Constant Rule:

$$\int kf(x)dx=k\int f(x)dx$$

3. Sum Rule:

$$\int [f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx$$

Students Frequently Make Mistakes

• Forgetting the constant of integration $+C$.

• Confusing integration with differentiation.

• Applying power rule incorrectly when $n=-1$.

• Errors in trigonometric integrals.

• Skipping modulus in $\ln \mathrm{\mid }x\mathrm{\mid }$.

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

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

$$\frac{x^3}{3}+C$$

Q2. Evaluate $\int x^{3}dx$.

$$\frac{x^4}{4}+C$$

Q3. Evaluate $\int \frac{1}{x}dx$.

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

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

$$e^x+C$$

Q5. Evaluate $\int \sin xdx$.

$$-\cos x+C$$

Q6. Evaluate $\int \cos xdx$.

$$\sin x+C$$

Q7. Evaluate $\int {\sec }^{2}xdx$.

$$\tan x+C$$

Q8. Evaluate $\int {\csc }^{2}xdx$.

$$-\cot x+C$$

Q9. Evaluate $\int \sec x\tan xdx$.

$$\sec x+C$$

Q10. Evaluate $\int \csc x\cot xdx$.

$$-\csc x+C$$

FAQs (10)

FAQ1. What is indefinite integral? Reverse process of differentiation.

FAQ2. Why add constant $C$? Because integration gives family of functions.

FAQ3. What is integral of $x^n$? $\frac{x^{n+1}}{n+1}+C,n\mathrm{\ne }-1$.

FAQ4. What is integral of $\frac{1}{x}$? $\ln \mathrm{\mid }x\mathrm{\mid }+C$.

FAQ5. What is integral of $e^x$? $e^x+C$.

FAQ6. What is integral of $\sin x$? $-\cos x+C$.

FAQ7. What is integral of $\cos x$? $\sin x+C$.

FAQ8. What is integral of ${\sec }^{2}x$? $\tan x+C$.

FAQ9. What is integral of ${\csc }^{2}x$? $-\cot x+C$.

FAQ10. Why is Exercise 7.1 important? It builds foundation for integrals and calculus applications.

Conclusion

Exercise 7.1 has 10 solved questions and 10 FAQs that strengthen your understanding of basic integration formulas. This begins the Integrals chapter in Class 12 Maths.

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