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Business Mathematics and Statistics · Class 12 Commerce

Ch 2Integral Calculus – I (Indefinite/Definite Integrals) — Class 12 Business Mathematics and Statistics, concept-first.

This Tamil Nadu HSC Class 12 Business Mathematics and Statistics chapter introduces integration — the reverse process of differentiation — building the toolkit that the next chapter uses for genuine business/economics applications like total cost, total revenue, and area under a demand curve.

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Key concepts

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Indefinite Integral and Standard Integrals

where ; standard forms include (), , , and the standard trigonometric integrals.

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CA Foundation

2 concepts · 3 appearances

2025, 2024

Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

1

Indefinite Integration — Standard Integrals

This Tamil Nadu HSC Class 12 Business Mathematics and Statistics chapter introduces integration — the reverse process of differentiation — building the toolkit that the next chapter uses for genuine b…

2

Integration by Substitution and by Parts

When the integrand is a function of a simpler expression (like ), substituting for that inner expression often reduces the integral to a standard form. If , then , and

3

Integration Using Partial Fractions

When the integrand is a rational function with factoring into distinct linear factors, the partial fraction decomposition rewrites it as a sum of simpler fractions that are each directly integrable us…

4

Definite Integrals — the Fundamental Theorem and Key Properties

The Fundamental Theorem of Calculus connects the indefinite integral (antiderivative) to a NUMBER — the definite integral over an interval :

Exercises

+Show 5 questions5 questions
  1. Q7Evaluate $\int\left(\sqrt{x}+\dfrac{1}{\sqrt{x}}\right)dx$.Free
  2. Q8Evaluate $\int(3x+2)^4\,dx$ using substitution.Free
  3. Q9Evaluate $\int x\ln x\,dx$ using integration by parts.Preview
  4. Q10Evaluate $\int\dfrac{dx}{x^2-1}$ using partial fractions.Preview
  5. Q11Evaluate $\displaystyle\int_1^2(3x^2-2x)\,dx$.Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 31 questions31 questions
  1. Q1If $\Gamma(n + 2) = 90\,\Gamma(n)$ ; $(n > 0)$, then the value of $'n'$ is : (a) $9$ (b) $8$ (c) $10$ (d) $7$Preview
  2. Q2$\int \dfrac{e^x}{\sqrt{1 + e^x}}\, dx$ is : (a) $2\sqrt{1 + e^x} + C$ (b) $e^x \sqrt{1 + e^x} + C$ (c) $\sqrt{1 + e^x} + C$ (d) $\dfrac{e^x…Preview
  3. Q3Evaluate : $\int \dfrac{\cos 2x + 2\sin^2 x}{\cos^2 x}\, dx$.Preview
  4. Q4Evaluate : $\int \dfrac{1}{\sqrt{x^2 + 6x + 13}}\, dx$.Preview
  5. Q5(a) Evaluate : $\int \dfrac{x}{2x^4 - 3x^2 - 2}\, dx$. OR (b) If the marginal cost of producing $x$ shoes is given by $(3xy + y^2)\, dx + (x…Preview
  6. Q6(a) Evaluate the integral as the limit of a sum $\int_1^2 (2x^3 - 4)\, dx$. OR (b) The following data relate to the life (in hours) of $6$ e…Preview
  7. Q7$\int_0^\infty e^{-2x}\, dx$ is ______. (a) $2$ (b) $0$ (c) $\frac{1}{2}$ (d) $1$Preview
  8. Q8If $f'(x) = 8x^3 - 2x$ and $f(2) = 8$, then find $f(x)$.Preview
  9. Q9Using second fundamental theorem, evaluate $\int_{-1}^{1} \frac{2x + 3}{x^2 + 3x + 7}\, dx$.Preview
  10. Q10(a) Integrate $x^3 e^{3x}$ with respect to $x$. OR (b) If 18% of the bolts produced by a machine are defective, determine the probability th…Preview
  11. Q11(a) Evaluate : $\int \frac{3x^2 + 6x + 1}{(x + 3)(x^2 + 1)}\, dx$. OR (b) Find an initial basic feasible solution of following problem using…Preview
  12. Q12The value of $\int \dfrac{\sin 2x}{2\sin x}\,dx$ is : (a) $\cos x+c$ (b) $\sin x+c$ (c) $\dfrac{1}{2}\cos x+c$ (d) $\dfrac{1}{2}\sin x+c$Preview
  13. Q13The value of $\int \sqrt{e^x}\,dx$ is : (a) $\dfrac{1}{2}\sqrt{e^x}+c$ (b) $\sqrt{e^x}+c$ (c) $\dfrac{1}{2\sqrt{e^x}}+c$ (d) $2\sqrt{e^x}+c$Preview
  14. Q14Evaluate : $\int (3+x)(2-5x)\,dx$.Preview
  15. Q15Evaluate : $\int \dfrac{1}{\sqrt{x+2}-\sqrt{x+3}}\,dx$Preview
  16. Q16(a) Evaluate the integral as the limit of a sum $\int_1^2 (2x+1)\,dx$ OR (b) Find the probability of guessing correctly atleast six of the t…Preview
  17. Q17The value of $\int \frac{\sin 2x}{2\cos x}\,dx$ is : (a) $\cos x+c$ (b) $\sin x+c$ (c) $-\sin x+c$ (d) $-\cos x+c$Preview
  18. Q18The value of $\Gamma(3/2)$ is : (a) $2\sqrt{\pi}$ (b) $\frac{3}{2}$ (c) $\frac{\sqrt{\pi}}{2}$ (d) $\sqrt{\pi}$Preview
  19. Q19Evaluate : $\int \sin^2 x\,dx$Preview
  20. Q20Evaluate : $\int \frac{1}{\sqrt{x+1}+\sqrt{x-1}}\,dx$Preview
  21. Q21(a) Evaluate : $\int \frac{4x^2+2x+6}{(x+1)^2(x-3)}\,dx$ OR (b) If the heights of 500 students are normally distributed with mean 68 inches…Preview
  22. Q22(a) Evaluate : $\int_{0}^{\frac{\pi}{2}} \frac{1}{1+\cot x}\,dx$ OR (b) Find a polynomial of degree two, which takes the values. | $x$ | 0 |…Preview
  23. Q23$\int \dfrac{\log x}{x}\,dx,\ (x>0)$ is : (a) $\dfrac{2}{x^2}+c$ (b) $\dfrac{1}{2}(\log x)^2+c$ (c) $-\dfrac{2}{x^2}+c$ (d) $-\dfrac{1}{2}(\…Preview
  24. Q24$\int_{0}^{4}\left(\sqrt{x}+\dfrac{1}{\sqrt{x}}\right)dx$ is : (a) $\dfrac{28}{3}$ (b) $\dfrac{20}{3}$ (c) $\dfrac{1}{3}$ (d) $\dfrac{21}{3}…Preview
  25. Q25Evaluate : $\int \dfrac{1}{\sin^{2}x\,\cos^{2}x}\,dx$Preview
  26. Q26Evaluate : $\int x\log x\,dx$Preview
  27. Q27The value of $\Gamma(10)$ is : (a) $9$ (b) $10$ (c) $9!$ (d) $10!$Preview
  28. Q28The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x \, dx$ is : (a) $1$ (b) $0$ (c) $4$ (d) $2$Preview
  29. Q29Evaluate $\int \frac{dx}{(2x + 3)^2}$Preview
  30. Q30Evaluate $\int \frac{x^2}{x^6 - 4} \, dx$Preview
  31. Q31(a) Evaluate : $\int_2^5 \frac{\sqrt{x}}{\sqrt{x} + \sqrt{7 - x}} \, dx$ OR (b) The population of a city in a census taken once in 10 years…Preview

More questions

+Show 6 questions6 questions
  1. Example 1Evaluate $\int(3x^2+4x+5)\,dx$.Free
  2. Example 2Evaluate $\int(2x+3)^5\,dx$ using substitution.Free
  3. Example 3Evaluate $\int xe^x\,dx$ using integration by parts.Preview
  4. Example 4Evaluate $\int\frac{dx}{(x+1)(x+2)}$ using partial fractions.Preview
  5. Example 5Evaluate $\displaystyle\int_1^3(2x+1)\,dx$.Preview
  6. Example 6Evaluate $\displaystyle\int_0^2 x^2\,dx$.Preview