MHT CET Integration: Essential Formulas and Worked Examples

To solve an integration question, first look for a standard form, an algebraic simplification or an inner function whose derivative is present. For an indefinite integral, include the constant of integration and check your result by differentiating it. For a definite integral, apply the limits consistently and remember that signed area can be negative.

These are original worked examples for revision, not claimed MHT CET past-paper questions. The guide covers selected core methods rather than the entire integration syllabus.

Which standard integrals should I know?

Using ∫ to denote integration, the power rule is ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C, for n ≠ −1 and on an appropriate domain. The excluded case is ∫(1/x) dx = ln|x| + C, on intervals that do not cross zero.

Other useful forms include ∫eˣ dx = eˣ + C, ∫cos x dx = sin x + C and ∫sin x dx = −cos x + C. Trigonometric arguments here use radians.

Do not memorise only the final expression. Differentiate each result once to see why it works and note its restrictions.

For example, the power rule cannot be used with n = −1 because it would require division by zero. That is a structural exception, not a small arithmetic detail.

When can I simplify before integrating?

Expand or rearrange when it turns the integrand into a sum of standard terms.

Example: ∫(3x² − 4x + 2) dx = x³ − 2x² + 2x + C. Integrate each term separately. Differentiating the answer gives 3x² − 4x + 2, which recovers the original integrand.

Linearity permits integrating a sum term by term. It does not permit replacing the integral of a product by a product of integrals, or the integral of a quotient by a quotient of integrals.

Before choosing a longer method, ask whether elementary algebra will expose familiar forms.

How do I recognise substitution?

Look for a composite expression accompanied by its derivative, possibly differing by a constant factor.

Example: For ∫2x cos(x²) dx, set u = x². Then du = 2x dx, so the integral becomes ∫cos u du = sin u + C = sin(x²) + C.

Differentiate sin(x²): the chain rule gives 2x cos(x²), confirming the result.

If the original integrand were x cos(x²), the factor would be one half: the result is one half of sin(x²) + C. Missing this multiplier is a common error. If no usable derivative factor is present, the same substitution may not simplify the problem.

How do definite integrals differ?

An indefinite integral represents a family of antiderivatives. A definite integral over stated limits gives a value.

Example: The integral of 2x from 0 to 2 is [x²] from 0 to 2 = 4. The arbitrary constant cancels when subtracting endpoint values.

If you substitute in a definite integral, either transform both limits into the new variable or return to the original variable before applying the original limits. Do not mix new-variable expressions with old-variable endpoints.

For a continuous function on an interval symmetric about zero, an odd function has integral zero. An even function has twice the integral over the positive half. Check both the symmetry and the interval before using those results.

Does a definite integral always equal geometric area?

No. A definite integral records signed accumulation. To calculate total geometric area between a curve and the x-axis, split at sign changes and add the nonnegative areas.

Example: The integral of x from −1 to 1 is zero: the negative and positive contributions cancel. The total area between the line and the x-axis on that interval is 1, the sum of two triangles each of area one half.

Confusing these questions can produce a perfectly calculated answer to the wrong task. Read whether the problem asks for an integral, signed area or total area.

Can I test myself with three short questions?

Try these original examples without looking at the answers:

1. Find an antiderivative of 6x².
2. Find an antiderivative of 3e³ˣ.
3. Evaluate the integral of cos x from 0 to π/2.

Answers: 1. 2x³ + C. 2. e³ˣ + C. 3. 1.

For the second answer, differentiating e³ˣ gives 3e³ˣ. For the third, evaluate sin x at the upper and lower limits. If a result is wrong, decide whether the cause was recall, chain-rule scaling or endpoint evaluation.

Integration FAQs

Is adding C always necessary?

Include it for an indefinite integral unless the problem supplies another convention or an initial condition fixes it. A definite integral evaluates to a value, so an arbitrary C is not added to that value.

Can I check every answer by differentiating?

Differentiation is a strong check for an antiderivative. For a definite integral, also check the endpoints, domain and any substitutions; a correct antiderivative can still be evaluated incorrectly.

Does this guide cover all integration methods?

No. It introduces standard forms, substitution and selected definite-integral checks. Use the official syllabus and your textbook for the full scope, including other prescribed techniques.

Explore Indefinite Integration, Definite Integration and the Maths preparation guide. For a mixed PCM baseline, start a free paper.