NCERT Solutions for Class 12 Mathematics Chapter 7: Integrals – Free PDF Download

Chapter 7 of Class 12 Maths is Integrals, the reverse process of differentiation. It covers indefinite and definite integrals, integration techniques, and the Fundamental Theorem of Calculus. These Class 12 Mathematics Chapter 7 solutions are also useful as quick revision notes before exams.

Integration as the Inverse of Differentiation

If d/dx[F(x)]=f(x), then ∫f(x)dx=F(x)+C, where C is the constant of integration. This is the indefinite integral.

Methods of Integration

Substitution method: change variables to simplify the integrand. Integration by partial fractions: used for rational functions, splitting into simpler fractions. Integration by parts: ∫u dv=uv−∫v du, useful for products of functions (using the ILATE rule to choose u).

Standard Integrals

Key results: ∫xⁿdx=xⁿ⁺¹/(n+1)+C (n≠−1); ∫1/x dx=log|x|+C; ∫e^xdx=e^x+C; ∫sin x dx=−cos x+C; ∫cos x dx=sin x+C; ∫1/√(a²−x²) dx=sin⁻¹(x/a)+C.

Definite Integrals

The definite integral ∫_a^bf(x)dx=F(b)−F(a), representing the (signed) area under the curve between x=a and x=b, evaluated via the Fundamental Theorem of Calculus.

Properties of Definite Integrals

Key properties used to simplify evaluation: ∫_a^bf(x)dx=−∫_b^af(x)dx; ∫_a^bf(x)dx=∫_a^bf(a+b−x)dx; ∫₀^2af(x)dx=2∫₀^af(x)dx if f(2a−x)=f(x), else 0.

Frequently Asked Questions

Why do we add a constant C in indefinite integration?
Because differentiation of a constant is zero, so infinitely many antiderivatives differ only by a constant.

What is the ILATE rule?
A priority order (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential) for choosing u in integration by parts.

What does a definite integral represent geometrically?
The signed area between the curve and the x-axis over the given interval.

Written by Satish

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