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Business Mathematics and Basic Statistics · Class 12 Commerce

Ch 14Integration — Class 12 Business Mathematics and Basic Statistics, concept-first.

Business Mathematics and Basic Statistics (WBCHSE Class 12 Commerce, Semester IV) now turns the Limits and Derivatives chapter's central operation around. Differentiation started from a function and found its rate of change.

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Q&A

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Concepts

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Key concepts

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Integration as the Inverse of Differentiation

If , then is an antiderivative of , and the indefinite integral collects the ENTIRE family of antiderivatives (differing by a constant ), since the derivative of any constant is zero.

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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.

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Integration as the Inverse of Differentiation

Business Mathematics and Basic Statistics (WBCHSE Class 12 Commerce, Semester IV) now turns the Limits and Derivatives chapter's central operation around.

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Standard Formula Integrals

This WBCHSE Class 12 Commerce Business Mathematics and Basic Statistics chapter fixes a small set of standard-formula integrals as the toolkit every worked example draws from.

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Integration by Parts — Algebraic Functions Only

Some products of two algebraic expressions — such as — are technically possible to integrate by fully expanding the power and integrating term by term, but that expansion quickly becomes long and erro…

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Definite Integration and the Fundamental Theorem of Integral Calculus

So far, every integral in this chapter has been an INDEFINITE integral — a family of functions, ending in .

Exercises

+Show 7 questions7 questions
  1. Q7Find $\displaystyle\int (x^{4}-3x^{2}+2)\,dx$ and verify your answer by differentiation.Free
  2. Q8Evaluate $\displaystyle\int (4x^{3}-6x+5)\,dx$.Free
  3. Q9Evaluate $\displaystyle\int \left(3e^{x} - \frac{4}{x}\right)dx$.Free
  4. Q10Evaluate $\displaystyle\int \frac{1}{x^{2}-25}\,dx$.Preview
  5. Q11Evaluate $\displaystyle\int \frac{1}{9-x^{2}}\,dx$.Preview
  6. Q12Evaluate $\displaystyle\int x(x-3)^{4}\,dx$ using integration by parts.Preview
  7. Q13Evaluate $\displaystyle\int_{0}^{3} (x^{2}+1)\,dx$ using the Fundamental Theorem of Integral Calculus.Preview

More questions

+Show 6 questions6 questions
  1. Example 1Verify that $F(x) = \dfrac{x^{3}}{3}+2x$ is an antiderivative of $f(x)=x^{2}+2$, and hence write down $\displaystyle\int (x^{2}+2)\,dx$.Free
  2. Example 2Evaluate $\displaystyle\int \left(3x^{4} - \frac{2}{x} + 5e^{x}\right)dx$.Free
  3. Example 3Evaluate $\displaystyle\int \frac{1}{x^{2}-9}\,dx$.Preview
  4. Example 4Evaluate $\displaystyle\int \frac{1}{4-x^{2}}\,dx$.Preview
  5. Example 5Evaluate $\displaystyle\int x(2x+1)^{3}\,dx$ using integration by parts.Preview
  6. Example 6State the Fundamental Theorem of Integral Calculus, and use it to evaluate $\displaystyle\int_{1}^{2}(3x^{2}+2x)\,dx$.Preview