Class 12 Maths Chapter 7 Integrals – Exercise 7.7 NCERT Solutions

Introduction

Exercise 7.7 focuses on definite integrals and their evaluation using properties and standard results. Students practice solving integrals involving trigonometric, exponential, and algebraic functions with limits. This exercise is crucial for mastering definite integrals and preparing for applications in area and probability.

Formulas Used

1. Fundamental Theorem of Calculus:

$${\int }_a^{b}f(x)dx=F(b)-F(a)$$

where $F(x)$ is antiderivative of $f(x)$.

2. Properties of Definite Integrals:

$${\int }_a^{b}f(x)dx=-{\int }_b^{a}f(x)dx$$

$${\int }_a^{b}f(x)dx={\int }_a^{c}f(x)dx+{\int }_c^{b}f(x)dx$$

$${\int }_0^{a}f(x)dx={\int }_0^{a}f(a-x)dx$$

3. Standard Results:

$${\int }_0^{\pi }\sin xdx=2,{\int }_0^{\pi }\cos xdx=0$$

$${\int }_0^1{x}^{n}dx=\frac{1}{n+1},n>-1$$

Students Frequently Make Mistakes

• Forgetting to apply properties before solving.

• Errors in substitution of limits.

• Confusing indefinite and definite integrals.

• Skipping symmetry property for simplification.

• Ignoring domain restrictions in trigonometric integrals.

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

Q1. Evaluate ${\int }_0^1{x}^{2}dx$.

$${\left[\frac{x^3}{3}\right]}_0^1=\frac{1}{3}$$

Q2. Evaluate ${\int }_0^1{x}^{3}dx$.

$${\left[\frac{x^4}{4}\right]}_0^1=\frac{1}{4}$$

Q3. Evaluate ${\int }_0^1(x+1)dx$.

$${[\frac{x^2}{2}+x]}_0^1=\frac{3}{2}$$

Q4. Evaluate ${\int }_0^{\pi }\sin xdx$.

$$[-\cos x{]}_0^{\pi }=2$$

Q5. Evaluate ${\int }_0^{\pi }\cos xdx$.

$$[\sin x{]}_0^{\pi }=0$$

Q6. Evaluate ${\int }_0^1{e}^{x}dx$.

$$[e^x{]}_0^1=e-1$$

Q7. Evaluate ${\int }_1^2\frac{1}{x}dx$.

$$[\ln x{]}_1^2=\ln 2$$

Q8. Evaluate ${\int }_0^1(x^2+1)dx$.

$${[\frac{x^3}{3}+x]}_0^1=\frac{4}{3}$$

Q9. Evaluate ${\int }_0^1(2x+3)dx$.

$$[x^2+3x{]}_0^1=4$$

Q10. Evaluate ${\int }_0^1\sqrt{x}dx$.

$${\left[\frac{2}{3}{x}^{3\mathrm{/}2}\right]}_0^1=\frac{2}{3}$$

FAQs (10)

FAQ1. What is definite integral? It represents area under curve between limits.

FAQ2. How is definite integral evaluated? By applying antiderivative at upper and lower limits.

FAQ3. What is difference between definite and indefinite integral? Definite has limits, indefinite has constant $+C$.

FAQ4. What is property of symmetry? ${\int }_0^{a}f(x)dx={\int }_0^{a}f(a-x)dx$.

FAQ5. What is ${\int }_a^{a}f(x)dx$? Zero.

FAQ6. What happens if limits reversed? Integral changes sign.

FAQ7. What is ${\int }_0^{\pi }\sin xdx$?

FAQ8. What is ${\int }_0^{\pi }\cos xdx$?

FAQ9. Why use definite integrals? To calculate area, displacement, probability.

FAQ10. Why is Exercise 7.7 important? It builds mastery of definite integrals and their properties.

Conclusion

Exercise 7.7 has 10 solved questions and 10 FAQs that strengthen your understanding of definite integrals and their properties. This completes the Integrals chapter in Class 12 Maths.

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