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Mathematics and Statistics · Class 12 Commerce

Ch 5Integration — Class 12 Mathematics and Statistics, concept-first.

In the earlier chapters of Maharashtra Board Std XII Commerce Mathematics and Statistics, differentiation took a function and produced its rate of change. Integration reverses that operation: it starts from a function and asks which function, when differentiated, gives this one back? This chapter builds the whole idea…

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Key concepts

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Standard Integrals and Rules

Integration is linear: a sum integrates term by term and constant factors pull outside the integral sign. The basic standard integrals are (), , , , and .

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In previous exams

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CA Foundation

2 concepts · 3 appearances

2025, 2024

MHT-CET

1 concept · 30 appearances

2026, 2025, 2024, 2023, 2022, 2021, 2020

Chapter contents

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

1

Integration as the Anti-derivative

In the earlier chapters of Maharashtra Board Std XII Commerce Mathematics and Statistics, differentiation took a function and produced its rate of change.

2

Standard Integrals and Rules of Integration

This section fixes the basic toolkit of standard integrals and the two rules that let them be combined.

3

Integration by Substitution

Many integrands are not standard forms as they stand, but become one after a change of variable. Integration by substitution replaces a chosen expression inside the integrand by a new variable , so th…

4

Integrals of Standard Forms

A group of frequently-occurring integrals involving have fixed results worth memorising. Each is derived once (by substitution or partial fractions) and then quoted directly.

5

Integration by Parts

When the integrand is a product of two functions that is not a substitution pattern — such as or — integration by parts is the technique to use.

6

Integration by Partial Fractions

A rational function — a ratio of two polynomials — often cannot be integrated directly, but can be split into a sum of simpler fractions that each ARE standard integrals.

Exercises

+Show 5 questions5 questions
  1. Q12Evaluate $\displaystyle\int \left(4x^{3} - \frac{6}{x} + 7\right)dx$.Free
  2. Q13Evaluate $\displaystyle\int 3x^{2}\,(x^{3}+4)^{4}\,dx$ using substitution.Free
  3. Q14Evaluate $\displaystyle\int \frac{dx}{9-x^{2}}$.Preview
  4. Q15Evaluate $\displaystyle\int x\,e^{2x}\,dx$ using integration by parts.Preview
  5. Q16Evaluate $\displaystyle\int \frac{dx}{x^{2}-5x+6}$ using partial fractions.Preview

Sample & Board Papers

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

+Show 23 questions23 questions
  1. Q1$\int \dfrac{1}{\sqrt{x^2 - 9}}\, dx =$ ______. (a) $\dfrac{1}{3} \log |x + \sqrt{x^2 - 9}| + c$ (b) $\log |x + \sqrt{x^2 - 9}| + c$ (c) $3\…Preview
  2. Q2State whether the following statement is true or false. If $\int \dfrac{4e^x - 25}{2e^x - 5}\, dx = Ax - 3 \log |2e^x - 5| + c$, where $c$ i…Preview
  3. Q3$\int \dfrac{x}{(x + 2)(x + 3)}\, dx =$ ______ $+ \int \dfrac{3}{x + 3}\, dx$Preview
  4. Q4Evaluate: $\int \dfrac{e^x}{\sqrt{e^{2x} + 4e^x + 13}}\, dx$Preview
  5. Q5$\int (1 - x)^{-2} \, dx = (1 - x)^{-1} + c$ (a) True (b) FalsePreview
  6. Q6Evaluate the following. $\int \frac{1}{7 + 6x - x^2} \, dx$Preview
  7. Q7Evaluate the following : $\int x^3 \cdot \log x \, dx$Preview
  8. Q8The value of $\displaystyle\int \dfrac{dx}{\sqrt{1 - x}}$ is ______. (a) $2\sqrt{1 - x} + c$ (b) $-2\sqrt{1 - x} + c$ (c) $\sqrt{x} + c$ (d)…Preview
  9. Q9$\displaystyle\int \dfrac{dx}{(x - 8)(x + 7)} =$ (a) $\dfrac{1}{15} \log \left|\dfrac{x + 2}{x - 1}\right| + c$ (b) $\dfrac{1}{15} \log \lef…Preview
  10. Q10For $\displaystyle\int \dfrac{x - 1}{(x + 1)^3}\, e^x\, dx = e^x f(x) + c$, $f(x) = (x + 1)^2$. (a) True (b) FalsePreview
  11. Q11If $f'(x) = \dfrac{1}{x} + x$ and $f(1) = \dfrac{5}{2}$, then $f(x) = \log x + \dfrac{x^2}{2} +$ ______Preview
  12. Q12If $f'(x) = 4x^3 - 3x^2 + 2x + k$, $f(0) = 1$ and $f(1) = 4$, find $f(x)$.Preview
  13. Q13Complete the following activity: $\displaystyle\int_0^2 \dfrac{dx}{4 + x - x^2}$ $= \displaystyle\int_0^2 \dfrac{dx}{-x^2 + \square + \squar…Preview
  14. Q14$\int \left(x + \frac{1}{x}\right)^3 dx$ = ______. (a) $\frac{1}{4}\left(x + \frac{1}{x}\right)^4 + c$ (b) $\frac{x^4}{4} + \frac{3x^2}{2} +…Preview
  15. Q15State whether the following statement is true or false: $\int \log x \, dx = x \log x + x + c$Preview
  16. Q16$\int e^x\left(\frac{1}{x} - \frac{1}{x^2}\right) dx$ = ______ + c.Preview
  17. Q17If $f'(x) = x^2 + 5$ and $f(0) = -1$ then $f(x)$ = ______.Preview
  18. Q18Evaluate the following. $\int \frac{1}{x(x^6 + 1)} \, dx$Preview
  19. Q19State whether the following statement is true or false: If $\int \frac{x}{(1 + x)(2 + x)}\,dx = \int \left(\frac{A}{1 + x} + \frac{B}{2 + x}…Preview
  20. Q20Evaluate: $\int \frac{1 + x}{x} + e^{-x}\,dx$Preview
  21. Q21Evaluate the following. $\int \frac{1}{4x^2 - 20x + 17}\,dx$Preview
  22. Q22Evaluate: $\displaystyle\int \dfrac{x^2}{x^4 + 5x^2 + 6}\, dx$Preview
  23. Q23Solve: $\displaystyle\int x^2 \sin x\, dx$Preview

More questions

+Show 11 questions11 questions
  1. Example 1Verify that $F(x)=\dfrac{x^{3}}{3}+4x$ is an anti-derivative of $f(x)=x^{2}+4$, and hence write down $\displaystyle\int (x^{2}+4)\,dx$.Free
  2. Example 2Evaluate $\displaystyle\int \left(x^{3} + \frac{3}{x} + 2e^{x} + 5^{x}\right)dx$.Free
  3. Example 3Evaluate $\displaystyle\int 2x\,(x^{2}+1)^{5}\,dx$ using substitution.Free
  4. Example 4Evaluate $\displaystyle\int \frac{2x+1}{x^{2}+x+7}\,dx$.Preview
  5. Example 5Evaluate $\displaystyle\int \frac{dx}{x^{2}-16}$.Preview
  6. Example 6Evaluate $\displaystyle\int \frac{dx}{x^{2}+9}$.Preview
  7. Example 7Evaluate $\displaystyle\int \frac{dx}{\sqrt{x^{2}+25}}$.Preview
  8. Example 8Evaluate $\displaystyle\int x\,e^{x}\,dx$ using integration by parts.Preview
  9. Example 9Evaluate $\displaystyle\int x\,\log x\,dx$ using integration by parts.Preview
  10. Example 10Evaluate $\displaystyle\int \frac{dx}{(x-1)(x-2)}$ using partial fractions.Preview
  11. Example 11Evaluate $\displaystyle\int \frac{x+3}{(x+1)(x-2)}\,dx$ using partial fractions.Preview