Class 12 Maths Chapter 7 Integrals – Exercise 7.5 NCERT Solutions

Introduction

Exercise 7.5 focuses on definite integrals as limits of sums. Students learn how to evaluate definite integrals using the definition, properties, and standard results. This exercise is crucial for understanding the fundamental theorem of calculus and applications in area problems.

Formulas Used

1. Definition of Definite Integral:

$${\int }_a^{b}f(x)dx=\underset{n\to \mathrm{\infty }}{\lim }\frac{b-a}{n}\sum _{i=1}^{n}f(a+i\frac{b-a}{n})$$

2. Basic Properties:

   • ${\int }_a^{a}f(x)dx=0$

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

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

3. Standard Results:

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

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

Students Frequently Make Mistakes

• Forgetting to apply properties of definite integrals.

• Confusing definite integrals with indefinite integrals.

• Errors in substitution of limits.

• Skipping symmetry property $f(x)$ vs $f(a-x)$.

• Ignoring negative sign when reversing limits.

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

Q1. Evaluate ${\int }_0^{1}xdx$.

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

Q2. Evaluate ${\int }_0^2{x}^{2}dx$.

$${\left[\frac{x^3}{3}\right]}_0^2=\frac{8}{3}$$

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 defined? As limit of sum of function values.

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 7.5 important? It builds foundation for definite integrals and applications.

Conclusion

Exercise 7.5 has 10 solved questions and 10 FAQs that strengthen your understanding of definite integrals as limits of sums. This builds the foundation for applications of definite integrals in Class 12 Maths.

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