Class 12 Maths Chapter 7 Integrals – Exercise 7.3 NCERT Solutions

Introduction

Exercise 7.3 focuses on integration using partial fractions. Students learn how to decompose rational functions into simpler fractions and integrate them easily. This technique is vital for solving complex integrals, especially in algebraic and calculus problems.

Formulas Used

1. Partial Fraction Decomposition: For rational function $\frac{P(x)}{Q(x)}$, where degree of $P(x)<$ degree of $Q(x)$,

$$\frac{P(x)}{Q(x)}=\frac{A}{(x-a)}+\frac{B}{(x-b)}+\dots$$

2. Integration of Simple Fractions:

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

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

Students Frequently Make Mistakes

• Forgetting to check degree condition before decomposition.

• Errors in solving coefficients $A,B,C$.

• Skipping modulus in logarithmic integrals.

• Confusing repeated linear factors with distinct ones.

• Forgetting constant of integration $+C$.

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

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

$$\frac{1}{x^2-1}=\frac{1}{2}(\frac{1}{x-1}-\frac{1}{x+1})$$

$$\int \frac{1}{x^2-1}dx=\frac{1}{2}\ln \mathrm{\mid }x-1\mathrm{\mid }-\frac{1}{2}\ln \mathrm{\mid }x+1\mathrm{\mid }+C$$

Q2. Evaluate $\int \frac{1}{x^2+3x+2}dx$.

$$x^2+3x+2=(x+1)(x+2)$$

$$\frac{1}{x^2+3x+2}=\frac{1}{x+1}-\frac{1}{x+2}$$

$$\int \frac{1}{x^2+3x+2}dx=\ln \mathrm{\mid }x+1\mathrm{\mid }-\ln \mathrm{\mid }x+2\mathrm{\mid }+C$$

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

$$\frac{x}{x^2-1}=\frac{1}{2}(\frac{1}{x-1}+\frac{1}{x+1})$$

$$\int \frac{x}{x^2-1}dx=\frac{1}{2}\ln \mathrm{\mid }x-1\mathrm{\mid }+\frac{1}{2}\ln \mathrm{\mid }x+1\mathrm{\mid }+C$$

Q4. Evaluate $\int \frac{2x+3}{x^2+3x+2}dx$. Split numerator:

$$\frac{2x+3}{(x+1)(x+2)}=\frac{1}{x+1}+\frac{1}{x+2}$$

$$\int \frac{2x+3}{x^2+3x+2}dx=\ln \mathrm{\mid }x+1\mathrm{\mid }+\ln \mathrm{\mid }x+2\mathrm{\mid }+C$$

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

$$\frac{1}{x(x+1)}=\frac{1}{x}-\frac{1}{x+1}$$

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

Q6. Evaluate $\int \frac{1}{x^2(x+1)}dx$.

$$\frac{1}{x^2(x+1)}=\frac{1}{x}-\frac{1}{x^2}-\frac{1}{x+1}$$

$$\int \frac{1}{x^2(x+1)}dx=\ln \mathrm{\mid }x\mathrm{\mid }+\frac{1}{x}-\ln \mathrm{\mid }x+1\mathrm{\mid }+C$$

Q7. Evaluate $\int \frac{x+2}{x^2+2x+1}dx$.

$$x^2+2x+1=(x+1{)}^2$$

$$\frac{x+2}{(x+1{)}^2}=\frac{1}{x+1}+\frac{1}{(x+1{)}^2}$$

$$\int \frac{x+2}{x^2+2x+1}dx=\ln \mathrm{\mid }x+1\mathrm{\mid }-\frac{1}{x+1}+C$$

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

$$x^2+x=x(x+1)$$

$$\frac{2x+1}{x(x+1)}=\frac{1}{x}+\frac{1}{x+1}$$

$$\int \frac{2x+1}{x^2+x}dx=\ln \mathrm{\mid }x\mathrm{\mid }+\ln \mathrm{\mid }x+1\mathrm{\mid }+C$$

Q9. Evaluate $\int \frac{3x+2}{x^2+2x}dx$.

$$x^2+2x=x(x+2)$$

$$\frac{3x+2}{x(x+2)}=\frac{1}{x}+\frac{2}{x+2}$$

$$\int \frac{3x+2}{x^2+2x}dx=\ln \mathrm{\mid }x\mathrm{\mid }+2\ln \mathrm{\mid }x+2\mathrm{\mid }+C$$

Q10. Evaluate $\int \frac{2x+3}{x^2+2x+1}dx$.

$$x^2+2x+1=(x+1{)}^2$$

$$\frac{2x+3}{(x+1{)}^2}=\frac{2}{x+1}+\frac{1}{(x+1{)}^2}$$

$$\int \frac{2x+3}{x^2+2x+1}dx=2\ln \mathrm{\mid }x+1\mathrm{\mid }-\frac{1}{x+1}+C$$

FAQs (10)

FAQ1. What is partial fraction decomposition? Breaking rational function into simpler fractions.

FAQ2. Why use partial fractions? To simplify integration.

FAQ3. What is condition for partial fractions? Degree of numerator < degree of denominator.

FAQ4. What if denominator has repeated factors? Include terms like $\frac{A}{(x-a)}+\frac{B}{(x-a{)}^2}$.

FAQ5. What is integral of $\frac{1}{x-a}$? $\ln \mathrm{\mid }x-a\mathrm{\mid }+C$.

FAQ6. Why modulus in logarithm? To cover both positive and negative values.

FAQ7. Can partial fractions be used for improper fractions? Yes, after long division.

FAQ8. What is advantage of partial fractions? Transforms complex integrals into standard forms.

FAQ9. Is constant of integration necessary? Yes, always add $+C$.

FAQ10. Why is Exercise 7.3 important? It builds foundation for advanced integration techniques.

Conclusion

Exercise 7.3 has 10 solved questions and 10 FAQs that strengthen your understanding of integration using partial fractions. This builds the foundation for advanced calculus in Class 12 Maths.

visit:www.fuzymathacademy.com