Lesson 13 of 13 · 16 min

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NCERT Wallah · Hinglish · Whole chapter · Open on YouTube

Must-know facts

19 facts

1∫f(x) dx = F(x) + C where F′ = f; any two antiderivatives differ by a constant.
2∫xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1; ∫dx/x = log|x| + C.
3∫aˣ dx = aˣ/log a + C; ∫eˣ dx = eˣ + C.
4∫tan x dx = log|sec x| + C; ∫sec x dx = log|sec x + tan x| + C.
5∫cot x dx = log|sin x| + C; ∫cosec x dx = log|cosec x − cot x| + C.
6∫f′(x)/f(x) dx = log|f(x)| + C.
7∫dx/(x² + a²) = (1/a) tan⁻¹(x/a) + C; ∫dx/√(a² − x²) = sin⁻¹(x/a) + C.
8∫dx/(x² − a²) = (1/2a) log|(x − a)/(x + a)| + C.
9∫dx/√(x² ± a²) = log|x + √(x² ± a²)| + C.
10Improper rational function: divide first, then split into partial fractions.
11By parts: ∫u v dx = u∫v dx − ∫(u′∫v dx) dx; ∫log x dx = x log x − x + C.
12∫eˣ[f(x) + f′(x)] dx = eˣ f(x) + C.
13∫eˣ sin x dx = (eˣ/2)(sin x − cos x) + C.
14A(x) = ∫ₐˣ f(t) dt has A′(x) = f(x) when f is continuous.
15∫ₐᵇ f(x) dx = F(b) − F(a); no + C in a definite integral.
16After a substitution in a definite integral, change the limits too.
17∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b − x) dx; ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx.
18Odd f on [−a, a] integrates to 0; even f gives 2∫₀ᵃ f dx.
19∫₀^(π/2) log sin x dx = −(π/2) log 2.

Common traps

Where marks are lost

Leaving out + C in an indefinite integral.

An indefinite integral is a family; write + C every time. Leave it out only in a definite integral, where it cancels.

Using xⁿ⁺¹/(n + 1) for n = −1.

That formula divides by zero. ∫dx/x = log|x| + C.

Writing ∫dx/x = log x without the modulus.

1/x lives on both sides of 0; log|x| covers both.

Integrating a product factor by factor: ∫f g dx = ∫f dx × ∫g dx.

There is no such rule. Use substitution, an identity or integration by parts.

Choosing the wrong first function in integration by parts, as in taking cos x first in ∫x cos x dx.

Take as first the factor that gets simpler on differentiating, here x.

Splitting an improper rational function straight into partial fractions.

If the degree of the numerator is not less than that of the denominator, divide first.

Keeping the old limits after substituting in a definite integral.

Either convert the limits to the new variable or return to x before putting in the limits.

Writing ∫₋₁² |x³ − x| dx as |∫₋₁² (x³ − x) dx|.

Split at every zero of x³ − x (−1, 0, 1) and remove the modulus piece by piece.

Using F(b) − F(a) across a point where f is undefined.

The second fundamental theorem needs f continuous on the whole of [a, b].

Formulas

12 to know

Power rule

∫xⁿ dx = xⁿ⁺¹/(n + 1) + C

n ≠ −1; for n = −1 the answer is log|x| + C.

Exponential

∫aˣ dx = aˣ/log a + C

a > 0, a ≠ 1; for a = e this is eˣ + C.

Tan and sec

∫tan x dx = log|sec x| + C; ∫sec x dx = log|sec x + tan x| + C

By substitution.

Sum of squares

∫dx/(x² + a²) = (1/a) tan⁻¹(x/a) + C

Complete the square first for a general quadratic.

Difference of squares

∫dx/(x² − a²) = (1/2a) log|(x − a)/(x + a)| + C

For a² − x² the fraction inside the log is (a + x)/(a − x).

Root forms

∫dx/√(a² − x²) = sin⁻¹(x/a) + C; ∫dx/√(x² ± a²) = log|x + √(x² ± a²)| + C

a > 0.

Integration by parts

∫u v dx = u∫v dx − ∫(u′ ∫v dx) dx

u is the first function, v the second.

eˣ pattern

∫eˣ[f(x) + f′(x)] dx = eˣ f(x) + C

Spot a function and its derivative.

Root of a² − x²

∫√(a² − x²) dx = (x/2)√(a² − x²) + (a²/2) sin⁻¹(x/a) + C

The x² ± a² forms use a log term instead.

Fundamental theorem

∫ₐᵇ f(x) dx = F(b) − F(a)

F′ = f, f continuous on [a, b].

Reflection property

∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b − x) dx

With a = 0: ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx.

Even and odd

∫₋ₐᵃ f dx = 2∫₀ᵃ f dx (f even); 0 (f odd)

Check f(−x) first.

Key terms

7 terms

Antiderivative: A function F whose derivative is the given f; also called a primitive.
Indefinite integral: The family F(x) + C of all antiderivatives of f.
Integrand: The function f(x) being integrated.
Constant of integration: The arbitrary real number C that shifts one antiderivative into another.
Proper rational function: P(x)/Q(x) with the degree of P less than the degree of Q.
Definite integral: ∫ₐᵇ f(x) dx, a single number; for f ≥ 0 the area under the curve from a to b.
Area function: A(x) = ∫ₐˣ f(t) dt, the area from a fixed a up to a moving x.