Integrals

Complete Practice Guide: Integrals Class 12

Integrals is a central Class 12 Mathematics chapter. Integration can be understood as the reverse process of differentiation and is also used to calculate accumulated quantities and areas. This worksheet provides repeated practice with standard forms, techniques of integration and definite integrals. The NCERT page supplied for this chapter is the official NCERT textbook portal, which identifies itself as “Textbooks PDF (I-XII)”. citeturn0view0

1. Meaning of Integration

If F′(x)=f(x), then an antiderivative of f(x) is F(x). The indefinite integral is written as ∫f(x)dx = F(x)+C, where C is the constant of integration. Always include C in an indefinite integral because many functions differ only by a constant but have the same derivative.

2. Basic Power Rule

For n≠−1, the basic rule is ∫xⁿdx = xⁿ⁺¹/(n+1)+C. Constants can be taken outside an integral: ∫k f(x)dx = k∫f(x)dx. The special case n=−1 gives ∫1/x dx = ln|x|+C. These formulas should be recalled quickly before moving to more complicated techniques.

3. Standard Integrals

Important standard results include ∫cos x dx = sin x+C, ∫sin x dx = −cos x+C, ∫sec²x dx = tan x+C, and ∫cosec²x dx = −cot x+C. Also remember ∫sec x tan x dx = sec x+C and ∫cosec x cot x dx = −cosec x+C. Standard forms reduce the amount of algebra required.

4. Substitution Method

Substitution is useful when an expression contains a function and its derivative. Put u=g(x) and then du=g′(x)dx. Rewrite the entire integral in terms of u, integrate, and substitute back. A good habit is to look for an inner expression whose derivative appears elsewhere in the integrand.

5. Integration by Parts

For a product of functions, use ∫u dv = uv − ∫v du. Choosing u wisely is important. A common guide is LIATE: logarithmic, inverse trigonometric, algebraic, trigonometric, exponential. It is not an absolute rule, but it is a useful starting point for selecting u.

6. Partial Fractions

Rational functions can sometimes be split into simpler fractions before integration. For example, a proper rational expression may be decomposed into terms such as A/(x−a) + B/(x−b). After decomposition, use standard logarithmic or power integrals. Check the decomposition before integrating to avoid carrying an algebraic error forward.

7. Trigonometric Integrals

Some trigonometric integrals require identities before applying standard formulas. Useful identities include sin²x+cos²x=1 and 1+tan²x=sec²x. Decide whether a substitution or identity will make the integral simpler. Keep powers and differentials organised carefully.

8. Definite Integrals

A definite integral has limits: ∫ₐᵇ f(x)dx = F(b)−F(a), where F′=f. There is no arbitrary +C in the final evaluation because the constant cancels. Definite integrals can represent signed area and accumulated change, so pay attention to the interval and the sign of the function.

9. Properties of Definite Integrals

Important properties include ∫ₐᵃ f(x)dx=0, ∫ₐᵇ f(x)dx=−∫ᵇₐ f(x)dx, and ∫ₐᵇ f(x)dx=∫ₐᶜ f(x)dx+∫ᶜᵇ f(x)dx. Symmetry can simplify integrals on intervals such as [−a,a]. Always check whether the function is even or odd before doing lengthy calculations.

10. Fundamental Theorem Idea

If F(x)=∫ₐˣ f(t)dt under suitable continuity conditions, then F′(x)=f(x). This connects differentiation and definite integration. It is especially useful in questions involving an integral whose upper limit is a variable.

11. Exam Strategy

First identify the structure of the integrand. Try a standard formula if it matches directly. If there is a composite expression with its derivative, try substitution. For products, consider integration by parts; for rational expressions, consider partial fractions. For definite integrals, use properties before expanding whenever possible. Differentiate your final antiderivative to verify an indefinite result.

12. Quick Formula Revision

• ∫xⁿdx=xⁿ⁺¹/(n+1)+C, n≠−1
• ∫1/x dx=ln|x|+C
• ∫u dv=uv−∫v du
• ∫ₐᵇf(x)dx=F(b)−F(a)
• ∫ₐᵃf(x)dx=0
• ∫ₐᵇf=−∫ᵇₐf
• F(x)=∫ₐˣf(t)dt ⇒ F′(x)=f(x)

13. How to Use This Worksheet

Choose a topic and difficulty, start a new test, attempt every question, and press Check Answer. Correct answers turn green and incorrect answers turn red. Show Answer stays locked until the question is checked. Use Shuffle Whole Worksheet for a new order, Timer for exam-style practice, and Save as PDF for offline revision. The responsive interface is designed for students using mobile phones as well as tablets and computers.