C
CraftExam
medium difficulty6 min

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.

In one line: Integration undoes differentiation, but the missing constant returns as +C.

Practice This Concept

Understand it

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.

Example

Since d/dx(x5/5)=x4, ∫x4 dx=x5/5+C.

Formula and rule

Key rules: ∫xn dx=xn+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.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Evaluate ∫(4x3-5) dx.

  1. 1Using linearity, ∫4x3 dx-∫5 dx=4·x4/4-5x+C=x4-5x+C.
  2. 2Final answer: x4-5x+C.

Method you can reuse

  1. 1Identify each term separately.
  2. 2Apply the standard integral formula.
  3. 3Keep constants outside the integral.
  4. 4Simplify coefficients.
  5. 5Add +C for an indefinite integral.
  6. 6Differentiate the result mentally to verify.

Visual tip

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.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Find ∫(3x2+2x) dx.

Think first, then show answer
x3+x2+C, because d/dx(x3+x2)=3x2+2x.

Common wrong answer

Writing ∫x4 dx=4x3+C by differentiating instead of integrating, or giving x3+x2 without +C for an indefinite integral.

Don't mix this up

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

Exam tip

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

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

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.
  1. 1

    Enough for 1 mark

    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

    Add for 2 marks

    Example: Since d/dx(x5/5)=x4, ∫x4 dx=x5/5+C.

  3. 3

    Add for 3 marks

    Integration finds a family of functions whose derivative is the given function.

MCQ Quiz

Practice this concept with focused MCQs

Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.

10 MCQs5 MinutesInstant Results
Practice This Concept

Help improve this page

Found something confusing, incorrect, or missing?

Practice This Concept