What this chapter is about
Integration is the reverse process of differentiation. Where differentiation breaks a function into its instantaneous rate of change, integration pieces together those rates to recover the original function or to find accumulated quantities such as area, volume or total distance. A Class 12 student meets this chapter because many problems in physics, economics and geometry require summing infinitely many infinitesimally small parts, and the integral is the precise tool for that task.
The chapter covers two main ideas: the indefinite integral (finding a family of functions whose derivative is a given function) and the definite integral (computing a number that represents net area under a curve between two limits). Students learn standard formulas, techniques such as substitution, integration by parts and partial fractions, and the link between the two kinds of integral through the Fundamental Theorem of Calculus. By the end, a student should be able to evaluate a wide variety of integrals and apply them to area problems.
Key ideas
- An antiderivative of f(x) is any function F(x) such that dF/dx = f(x). The indefinite integral ∫ f(x) dx represents the family of all antiderivatives, written F(x) + C, where C is an arbitrary constant.
- Standard results for powers, exponentials, trigonometric and inverse trigonometric functions form the foundation: for instance, ∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C when n ≠ −1.
- Substitution (change of variable) simplifies an integral by replacing a complicated inner function with a single new variable.
- Integration by parts follows from the product rule: ∫ u dv = uv − ∫ v du, useful when the integrand is a product of two different types of function.
- Partial fractions break a rational function into simpler pieces that can each be integrated using logarithms or inverse-tangent formulas.
- The definite integral ∫ from a to b of f(x) dx equals F(b) − F(a), where F is any antiderivative of f. This is the Fundamental Theorem of Calculus.
- Geometrically, the definite integral gives the net signed area between the curve y = f(x) and the x-axis from x = a to x = b.
- Properties such as ∫ from a to b = −∫ from b to a, and ∫ from a to b + ∫ from b to c = ∫ from a to c, help evaluate and manipulate definite integrals.
Formulas and facts to remember
- Formula: ∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C, n ≠ −1 · Meaning: Power rule for integration
- Formula: ∫ (1/x) dx = ln |x| + C · Meaning: Integral of reciprocal gives natural log
- Formula: ∫ eˣ dx = eˣ + C · Meaning: Exponential unchanged on integration
- Formula: ∫ aˣ dx = aˣ / ln a + C, a > 0, a ≠ 1 · Meaning: General exponential base a
- Formula: ∫ sin x dx = −cos x + C · Meaning: Sine integrates to negative cosine
- Formula: ∫ cos x dx = sin x + C · Meaning: Cosine integrates to sine
- Formula: ∫ sec²x dx = tan x + C · Meaning: Derivative of tan x is sec²x
- Formula: ∫ 1 / (1 + x²) dx = arctan x + C · Meaning: Leads to inverse tangent
- Formula: ∫ 1 / √(1 − x²) dx = arcsin x + C · Meaning: Leads to inverse sine
- Formula: ∫ u dv = uv − ∫ v du · Meaning: Integration by parts rule
- Formula: ∫ from a to b of f(x) dx = F(b) − F(a) · Meaning: Fundamental Theorem of Calculus
Worked examples
Example 1 — Power and sum rule
Evaluate ∫ (3x⁴ − 2x + 5) dx.
Step 1: Integrate term by term. ∫ 3x⁴ dx = 3 × x⁵ / 5 = (3/5)x⁵. ∫ −2x dx = −2 × x² / 2 = −x². ∫ 5 dx = 5x.
Step 2: Combine and add the constant. Answer: (3/5)x⁵ − x² + 5x + C.
Example 2 — Substitution
Evaluate ∫ 2x cos(x²) dx.
Step 1: Let u = x², so du/dx = 2x, hence du = 2x dx.
Step 2: Rewrite the integral in terms of u. ∫ cos u du = sin u + C.
Step 3: Substitute back. Answer: sin(x²) + C.
Example 3 — Integration by parts
Evaluate ∫ x eˣ dx.
Step 1: Choose u = x (algebraic, easy to differentiate) and dv = eˣ dx (easy to integrate). Then du = dx and v = eˣ.
Step 2: Apply the formula. ∫ x eˣ dx = x eˣ − ∫ eˣ dx = x eˣ − eˣ + C.
Step 3: Factor if desired. Answer: eˣ (x − 1) + C.
Common mistakes
Forgetting the constant C in indefinite integrals → always add + C because any constant vanishes on differentiation.
Applying the power rule when n = −1 → use ∫ (1/x) dx = ln |x| + C instead.
Dropping the differential du or dx during substitution → keep track; the differential tells you how to replace dx with du.
Wrong choice of u in integration by parts, leading to a harder integral → try LIATE priority: Logarithm, Inverse trig, Algebraic, Trigonometric, Exponential; pick u from earlier in this list.
Sign errors when evaluating F(b) − F(a) → substitute upper limit first, then subtract the value at the lower limit.
Quick revision
- Integration undoes differentiation; add C for the general antiderivative.
- Master the standard table: powers, 1/x, eˣ, sin x, cos x, sec²x, 1/(1 + x²), 1/√(1 − x²).
- Substitution replaces a composite function with a simpler variable.
- Integration by parts: ∫ u dv = uv − ∫ v du; use LIATE to pick u.
- Fundamental Theorem: ∫ from a to b of f(x) dx = F(b) − F(a).