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Integration as the Reverse of Differentiation

If F'(x)=f(x), then the indefinite integral of f(x) is written as ∫f(x) dx=F(x)+C, where C is an arbitrary constant.

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Student-friendly explanation

Integration finds a family of functions whose derivative is the given function. The constant C is essential because many functions differing only by a constant have the same derivative. Direct integration questions usually test recognition of standard derivative-integral pairs and correct use of linearity.

How to write this in exams

  1. 1

    Start with the exact idea

    If F'(x)=f(x), then the indefinite integral of f(x) is written as ∫f(x) dx=F(x)+C, where C is an arbitrary constant.

  2. 2

    Then show how to use it

    Identify each term separately. Apply the standard integral formula. Keep constants outside the integral. Simplify coefficients. Add +C for an indefinite integral. Differentiate the result mentally to verify.

  3. 3

    Add one concrete example

    Since d/dx(x^5/5)=x^4, ∫x^4 dx=x^5/5+C.

  4. 4

    Avoid this incomplete answer

    Writing ∫x^4 dx=4x^3+C by differentiating instead of integrating, or giving x^3+x^2 without +C for an indefinite integral.

Definition

If F'(x)=f(x), then the indefinite integral of f(x) is written as ∫f(x) dx=F(x)+C, where C is an arbitrary constant.

Example

Since d/dx(x^5/5)=x^4, ∫x^4 dx=x^5/5+C.

Rule to remember

Key rules: ∫x^n dx=x^(n+1)/(n+1)+C for n≠-1; ∫kf(x) dx=k∫f(x) dx; ∫[f(x)+g(x)] dx=∫f(x) dx+∫g(x) dx. The constant C is required for indefinite integrals.

Memory hook

Integration undoes differentiation, but the missing constant returns as +C.

Examples and method

Worked example

Evaluate ∫(4x^3-5) dx. Using linearity, ∫4x^3 dx-∫5 dx=4·x^4/4-5x+C=x^4-5x+C. Final answer: x^4-5x+C.

Method to apply

Identify each term separately. Apply the standard integral formula. Keep constants outside the integral. Simplify coefficients. Add +C for an indefinite integral. Differentiate the result mentally to verify.

Diagram support

A graph is not required for routine antiderivative questions. The useful idea is that all antiderivatives differ by a vertical shift, so their derivatives remain the same.

How CBSE asks it

Usually asked as a direct 1-mark or 2-mark question, sometimes mixed with standard integrals of powers, exponentials, trigonometric functions, and simple algebraic sums.

Avoid common mistakes

Common confusion

Students often omit +C in indefinite integrals or divide by the old power incorrectly in ∫x^n dx.

Common wrong answer

Writing ∫x^4 dx=4x^3+C by differentiating instead of integrating, or giving x^3+x^2 without +C for an indefinite integral.

Exam tip

For every indefinite integral, check your answer by differentiating it. If the derivative gives the integrand, the integration is correct.

Quick check

Find ∫(3x^2+2x) dx.

x^3+x^2+C, because d/dx(x^3+x^2)=3x^2+2x.

Answer writing and exam use

1-mark answer

If F'(x)=f(x), then the indefinite integral of f(x) is written as ∫f(x) dx=F(x)+C, where C is an arbitrary constant.

2-mark answer

If F'(x)=f(x), then the indefinite integral of f(x) is written as ∫f(x) dx=F(x)+C, where C is an arbitrary constant. Key rules: ∫x^n dx=x^(n+1)/(n+1)+C for n≠-1; ∫kf(x) dx=k∫f(x) dx; ∫[f(x)+g(x)] dx=∫f(x) dx+∫g(x) dx. The constant C is required for indefinite integrals. Since d/dx(x^5/5)=x^4, ∫x^4 dx=x^5/5+C.

3-mark answer

Integration finds a family of functions whose derivative is the given function. The constant C is essential because many functions differing only by a constant have the same derivative. Direct integration questions usually test recognition of standard derivative-integral pairs and correct use of linearity. Key rules: ∫x^n dx=x^(n+1)/(n+1)+C for n≠-1; ∫kf(x) dx=k∫f(x) dx; ∫[f(x)+g(x)] dx=∫f(x) dx+∫g(x) dx. The constant C is required for indefinite integrals. Evaluate ∫(4x^3-5) dx. Using linearity, ∫4x^3 dx-∫5 dx=4·x^4/4-5x+C=x^4-5x+C. Final answer: x^4-5x+C. Usually asked as a direct 1-mark or 2-mark question, sometimes mixed with standard integrals of powers, exponentials, trigonometric functions, and simple algebraic sums. Writing ∫x^4 dx=4x^3+C by differentiating instead of integrating, or giving x^3+x^2 without +C for an indefinite integral.
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