Class 12 Mathematics Chapter 7 Integrals – Revision Notes

Integrals reverses the process of differentiation, and this chapter collects the standard results along with techniques like substitution, partial fractions, and integration by parts needed to evaluate them.

Last Updated: September 23, 2026

Standard Integrals

• ∫xⁿdx=xⁿ⁺¹/(n+1)+C; ∫1/x dx=log|x|+C; ∫e^xdx=e^x+C.
• ∫sin x dx=−cos x+C; ∫cos x dx=sin x+C.

Techniques

• Substitution, partial fractions, integration by parts (∫u dv=uv−∫v du, ILATE order).

Definite Integrals

• ∫_a^bf(x)dx=F(b)−F(a) (Fundamental Theorem).
• Key property: ∫_a^bf(x)dx=∫_a^bf(a+b−x)dx.

One-Line Summary

Integration reverses differentiation; indefinite integrals carry a constant C, while definite integrals compute exact signed area using the Fundamental Theorem of Calculus and simplifying properties.

Quick visual: a worked diagram from the full Extra Questions page, for reference.

Frequently Asked Questions

What is the relationship between integration and differentiation?
Integration is the reverse process of differentiation; finding the integral of a function means finding another function whose derivative gives back the original function, which is why integration is also called antidifferentiation.

What is the difference between definite and indefinite integrals?
An indefinite integral gives a general family of antiderivative functions with an added constant of integration, while a definite integral is evaluated between two limits and gives a specific numerical value, often representing area under a curve.

Written by Satish

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