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Mathematics and Statistics · Class 12 Commerce

Ch 6Definite Integration — Class 12 Mathematics and Statistics, concept-first.

In the previous chapter, integration reversed differentiation: given a function , its indefinite integral was itself a function (an antiderivative, plus an arbitrary constant ).

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Key concepts

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Definite Integral and the Fundamental Theorem

The definite integral evaluates to a number. By the Fundamental Theorem of Calculus, if then ; the arbitrary constant cancels on subtraction. The result is the net signed area between and the -axis over .

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Chapter contents

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

1

The Definite Integral and the Fundamental Theorem

In the previous chapter, integration reversed differentiation: given a function , its indefinite integral was itself a function (an antiderivative, plus an arbitrary constant ).

2

Standard Definite Integrals and Linearity

Evaluating a definite integral needs the same list of standard antiderivatives used for indefinite integration, now applied between limits.

3

Properties of Definite Integrals

Definite integrals obey several properties that often make evaluation far shorter than direct antidifferentiation. Each is stated below with the situation it is used in.

4

Evaluation by Substitution

When the integrand is a composite function times (a constant multiple of) the derivative of the inner function, the substitution method simplifies it.

5

Even and Odd Functions over Symmetric Limits

When the limits of integration are symmetric about — that is, of the form to — the symmetry of the integrand can be exploited to shorten or immediately settle the integral.

Exercises

+Show 5 questions5 questions
  1. Q10Evaluate $\displaystyle\int_{2}^{4} (x + 3)\,dx$.Free
  2. Q11Evaluate $\displaystyle\int_{1}^{2} (x^{2} + 2x)\,dx$.Free
  3. Q12Evaluate $\displaystyle\int_{1}^{3} \frac{1}{x}\,dx$.Preview
  4. Q13Evaluate $\displaystyle\int_{0}^{1} 3x^{2}\,e^{x^{3}}\,dx$ using substitution.Preview
  5. Q14Evaluate $\displaystyle\int_{0}^{1} \frac{x}{x^{2} + 1}\,dx$ using substitution.Preview

Sample & Board Papers

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

+Show 11 questions11 questions
  1. Q1$\int_{-5}^{5} \dfrac{x^7}{x^4 + 10}\, dx =$ ______. (a) 10 (b) 5 (c) 0 (d) $\dfrac{1}{5}$Preview
  2. Q2Evaluate the following definite integral: $\int_1^3 \log x\, dx$Preview
  3. Q3$\int_0^2 e^x \, dx$ = ______. (a) $e^2 - 1$ (b) $1 - e^2$ (c) $e - 1$ (d) $1 - e$Preview
  4. Q4Evaluate: $\int_1^3 \frac{\sqrt{x + 5}}{\sqrt{x + 5} + \sqrt{9 - x}} \, dx$Preview
  5. Q5$\displaystyle\int_a^b f(x)\, dx = \int_a^b f(t)\, dt$ (a) True (b) FalsePreview
  6. Q6$\int_2^7 \frac{\sqrt{x}}{\sqrt{x} + \sqrt{9 - x}} \, dx$ = ______. (a) $\frac{7}{2}$ (b) $\frac{5}{2}$ (c) 7 (d) 2Preview
  7. Q7Evaluate: $\int_1^2 \frac{1}{x^2 + 6x + 5} \, dx$Preview
  8. Q8If $\int_0^a 3x^2\,dx = 8$ then $a$ = ______. (a) 2 (b) 0 (c) $\frac{8}{3}$ (d) 1Preview
  9. Q9Evaluate the following integrals: $\int_2^7 \frac{\sqrt{x}}{\sqrt{x} + \sqrt{9 - x}}\,dx$Preview
  10. Q10Evaluate: $\displaystyle\int_0^2 \dfrac{1}{\sqrt{4 - x^2}}\, dx$Preview
  11. Q11Evaluate: $\displaystyle\int_3^9 \dfrac{\sqrt[3]{12 - x}}{\sqrt[3]{x} + \sqrt[3]{12 - x}}\, dx$Preview

More questions

+Show 9 questions9 questions
  1. Example 1Evaluate $\displaystyle\int_{1}^{3} (2x + 1)\,dx$.Free
  2. Example 2Evaluate $\displaystyle\int_{0}^{2} (3x^{2} - 2x + 1)\,dx$.Free
  3. Example 3Evaluate $\displaystyle\int_{1}^{2} \frac{1}{x}\,dx$.Free
  4. Example 4Evaluate $\displaystyle\int_{0}^{1} e^{x}\,dx$.Preview
  5. Example 5If $\displaystyle\int_{0}^{2} f(x)\,dx = 5$ and $\displaystyle\int_{2}^{6} f(x)\,dx = 3$, find (i) $\displaystyle\int_{0}^{6} f(x)\,dx$ and…Preview
  6. Example 6Evaluate $\displaystyle\int_{0}^{1} 2x\,(x^{2}+1)^{3}\,dx$ using substitution.Preview
  7. Example 7Evaluate $\displaystyle\int_{1}^{e} \frac{\log x}{x}\,dx$ using substitution.Preview
  8. Example 8Evaluate $\displaystyle I = \int_{0}^{4} \frac{\sqrt{x}}{\sqrt{x} + \sqrt{4 - x}}\,dx$ using a property of definite integrals.Preview
  9. Example 9Evaluate $\displaystyle\int_{-1}^{1} (x^{2} + 1)\,dx$ using the even/odd function property, and state the value of $\displaystyle\int_{-2}^{…Preview