Business Mathematics and Statistics · Class 12 Commerce
Ch 7Integration — Class 12 Business Mathematics and Statistics, concept-first.
The CHSE Odisha (Council of Higher Secondary Education) Class 11 Commerce syllabus places Integration directly after Differentiation in the Business Mathematics and Statistics elective — and for good reason: integration is differentiation run in reverse.
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.
Most relevant Q&A
In previous exams
How often this chapter’s concepts have been examined — real appearance data, never estimated.
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Integration as the Inverse of Differentiation
The CHSE Odisha (Council of Higher Secondary Education) Class 11 Commerce syllabus places Integration directly after Differentiation in the Business Mathematics and Statistics elective — and for good…
Standard Integrals
This chapter fixes a small set of standard integrals as the toolkit every later worked example draws from — each one provable, and checkable, by differentiating the right-hand side.
Integration by Substitution
Some integrands are not directly one of the standard forms, but become one after a change of variable.
Integration by Parts
Some products cannot be integrated by any substitution because neither factor is (a constant multiple of) the derivative of the expression inside the other — a plain polynomial-times-exponential produ…
Definite Integrals and Their Properties
Every integral so far in this chapter has been an INDEFINITE integral — a family of functions, ending in .
Business Applications of Integration
Because integration reverses differentiation, and because marginal cost and marginal revenue are THEMSELVES derivatives (of total cost and total revenue respectively — the subject of the earlier Diffe…
Exercises
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- Q9Find $\displaystyle\int (x^{5}+4x-7)\,dx$ and verify your answer by differentiation.Free
- Q10Evaluate $\displaystyle\int \left(4x^{3} - \frac{5}{x} + 3e^{x}\right)dx$.Free
- Q11Evaluate $\displaystyle\int 4x(2x^{2}-3)^{3}\,dx$ using substitution.Preview
- Q12Evaluate $\displaystyle\int_{1}^{4} (2x+1)\,dx$ directly, and again by splitting the interval at $x=2$ using the additivity property. Confir…Preview
- Q13The demand function for a commodity is $p=20-2x$ and the supply function is $p=2+x$ (where $x$ is quantity and $p$ is price in rupees). Find…Preview
- Q14If $\displaystyle\int_{2}^{7} f(x)\,dx = 15$, what is the value of $\displaystyle\int_{7}^{2} f(x)\,dx$? (a) $15$ (b) $-15$ (c) $0$ (d) $30$Preview
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
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- Q1Answer each of the following questions in one sentence: (xii) Integrate $3x^2 + 1$ with respect to $x$.Preview
- Q2(c) $\int 2x \, dx$ is equal to (a) $x^2$ (b) $2$ (c) $\dfrac{1}{x^2}$ (d) $\dfrac{2}{x^2}$Preview
- Q3(j) Evaluate: $\int (8x^3 + 33x^2 - 6x - 7) \, dx$.Preview
- Q4$\int x^5 \, dx$ is equal to : (a) $5x^4 + C$ (b) $4x^5$ (c) $\dfrac{x^6}{6}$ (d) $\dfrac{x^6}{6} + C$Preview
- Q5Evaluate : $\int \dfrac{1}{x^4} \, dx$Preview
- Q6The integral of $(x^2 + 4)^2$ with respect to x, is : (a) $\frac{x^5}{5} + \frac{8}{3}x^3 + 16x + c$ (b) $\frac{x^3}{3} + 8x^2 + 16x + c$ (c…Preview
- Q7Fill in the blanks : The integral of $(2x + 4)^5$ with respect to x is ________ + c.Preview
- Q8Evaluate : $\int \frac{3}{4x^2}\,dx$Preview
More questions
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- Example 1Verify that $F(x) = \dfrac{x^{4}}{4} - 3x$ is an antiderivative of $f(x)=x^{3}-3$, and hence write down $\displaystyle\int (x^{3}-3)\,dx$.Free
- Example 2Evaluate $\displaystyle\int \left(5x^{4} - \frac{3}{x} + 2e^{x}\right)dx$.Free
- Example 3Evaluate $\displaystyle\int 3x^{2}(x^{3}+4)^{4}\,dx$ using substitution.Free
- Example 4Evaluate $\displaystyle\int x\,e^{x}\,dx$ using integration by parts.Preview
- Example 5Evaluate $\displaystyle\int_{0}^{2} (3x^{2}+4)\,dx$ using the Fundamental Theorem of Integral Calculus.Preview
- Example 6The marginal cost of a firm is $MC(x) = 6x+4$ (in rupees per unit) and the fixed cost is ₹100. Find the total cost function $TC(x)$, and hen…Preview
- Example 7The marginal revenue function of a firm is $MR(x) = 50-4x$. Find the total revenue function $TR(x)$ and the demand function $p(x)$.Preview
- Example 8Find the area bounded by the curve $y=x^{2}+2$, the $x$-axis, and the ordinates $x=1$ and $x=3$.Preview