IGCSE, Edexcel, IB and AP Science Courses | Dr Dabour Hub Academy

Overview

Treat indefinite integration as the reverse process of differentiation, with its properties and the table of standard integrals.
Integrate exponential, logarithmic and trigonometric functions and use them in geometric and physical applications.
Use differentials, integration by substitution and integration by parts, and recover a curve from the slope of its tangent.

Indefinite Integration of Trigonometric, Exponential and Logarithmic Functions
├── Antiderivatives and the indefinite integral
│   ├── Definition of the antiderivative
│   ├── Properties of the indefinite integral
│   └── Standard (tabular) integrals
├── Integrals of exponential functions
├── Integrals of logarithmic functions
│   ├── The function 1/x
│   └── Multiples of the logarithmic function
├── Applications of indefinite integration
│   ├── Geometric applications
│   └── Physical applications
├── Differentials
├── Integration by substitution
└── Integration by parts (LIATE)

Indefinite Integration of Trigonometric, Exponential and Logarithmic Functions · 1 / 11

The antiderivative and the indefinite integral

Definition

An antiderivative (inverse derivative) of f(x) on an interval is a function F(x) whose derivative is f(x), that is F′(x) = f(x) for every x in the interval.
Adding any constant to an antiderivative gives another antiderivative, so every function has infinitely many antiderivatives.
The set of all antiderivatives is called the indefinite integral and is written ∫ f(x) dx: ∫ f(x) dx = F(x) + C.
Geometrically the indefinite integral is a family of parallel curves differing only by the constant, so at the same value of x their tangents all have the same slope.
C is the constant of integration (arbitrary constant); it is fixed when a point on the curve is known.

Indefinite Integration of Trigonometric, Exponential: three parallel curved wires y = x² + C show indefinite integration and the integrals of exponential functions with slope 2 at x = 1.

Properties

Integral of a sum or difference∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx.

Constant factor∫ k f(x) dx = k ∫ f(x) dx, where k is a fixed real number.

Power rule∫ xn dx = xn+1/(n + 1) + C, n ≠ −1. The excluded case is exactly the logarithmic one: ∫ dx/x = ln|x| + C.

Check any result by differentiatingd/dx of the answer must return the integrand.

∫ (3x² + 4x + 5) dx = x³ + 2x² + 5x + C.
∫ (x² + 2x − 3) dx = x³/3 + x² − 3x + C.
∫ (√x + 1/√x) dx = ∫ (x^1/2 + x^−1/2) dx = (2/3)x^3/2 + 2x^1/2 + C.
substitution is the efficient route (section 07) — u = x² − 3, du = 2x dx → ½∫ u² du = (x² − 3)³/6 + C. Expanding first also works: ∫ (x⁵ − 6x³ + 9x) dx = x⁶/6 − 3x⁴/2 + 9x²/2 + C, which differs from the factored form only by a constant.

Indefinite Integration of Trigonometric, Exponential and Logarithmic Functions · 2 / 11

The table of standard integrals (including trigonometric)

Integrand	Integral	Comes from
cos x	sin x + C	d/dx(sin x) = cos x
sin x	−cos x + C	d/dx(−cos x) = sin x
sec² x	tan x + C	d/dx(tan x) = sec² x
cosec² x	−cot x + C	d/dx(−cot x) = cosec² x
sec x tan x	sec x + C	d/dx(sec x) = sec x tan x
cosec x cot x	−cosec x + C	d/dx(−cosec x) = cosec x cot x
tan x	ln\|sec x\| + C	write tan x = sin x/cos x, u = cos x
cot x	ln\|cosec x\| + C	write cot x = cos x/sin x, u = sin x
e^x	e^x + C	self-derivative
1/x	ln\|x\| + C	d/dx ln\|x\| = 1/x

Whenever the angle is (ax + b), divide by a ∫ cos(ax + b) dx = sin(ax + b)/a + C, ∫ sin(ax + b) dx = −cos(ax + b)/a + C.
∫ sec²(2x) dx = tan(2x)/2 + C and ∫ e^3x dx = e^3x/3 + C — the same rule.
Squares of trig functions are reduced with the double-angle identities first: sin²x = (1 − cos 2x)/2, cos²x = (1 + cos 2x)/2.
Example ∫ sin²x dx = ∫ (1 − cos 2x)/2 dx = x/2 − sin 2x/4 + C.

Indefinite Integration of Trigonometric, Exponential and Logarithmic Functions · 3 / 11

Integrals of exponential functions

The two rules

Comparison aspect	Main idea	Keywords
Natural base	∫ eu du = eu + C so for a linear exponent ∫ eax+b dx = eax+b/a + C	linearexponent
Any positive base a ≠ 1	∫ au du = au/ln a + C so ∫ akx dx = akx/(k ln a) + C
Reason	d/dx(ax) = ax ln a therefore the factor ln a must be divided out when integrating	thereforefactormustdivided
•	If the exponent is a function g(x), the chain rule must be undone: ∫ e^g(x) g′(x) dx = e^g(x) + C.	exponentfunctionchainrule

∫ e^{2x − 1} dx = e^{2x − 1}/2 + C.
∫ 3^x dx = 3^x/ln 3 + C.
let u = x², du = 2x dx → ½∫ eu du = ex²/2 + C.
∫ (e^x + 2^x) dx = e^x + 2^x/ln 2 + C — integrate term by term, different rules for each.

Indefinite Integration of Trigonometric, Exponential and Logarithmic Functions · 4 / 11

Integrals of logarithmic functions (the 1/x rule)

The rule

∫ dx/x = ln|x| + C.The modulus is compulsory: for x < 0, d/dx ln(−x) = (−1)/(−x) = 1/x, so ln|x| is an antiderivative on both sides of zero.

Linear argument∫ dx/(ax + b) = ln|ax + b|/a + C.

Numerator equal to the derivative of the denominator∫ f′(x)/f(x) dx = ln|f(x)| + C.

Multiples and powers∫ k·dx/x = k ln|x| + C = ln|x|k + C, and ∫ dx/x² = −1/x + C (a power, not a logarithm).

∫ dx/(3x − 5) = ln|3x − 5|/3 + C.
numerator is the derivative of the denominator → ln(x² + x + 4) + C (the quadratic is always positive, so the modulus may be dropped).
∫ dx/x³ = ∫ x⁻³ dx = x⁻²/(−2) + C = −1/(2x²) + C. Do not use the log rule unless the power of x in the denominator is exactly one.
∫ tan x dx = ∫ sin x/cos x dx = −ln|cos x| + C = ln|sec x| + C.
∫ ln x dx = x ln x − x + C (by parts, section 08).

Indefinite Integration of Trigonometric, Exponential and Logarithmic Functions · 5 / 11

Applications of the indefinite integral

Geometric: a curve from its slope

If a curve y = f(x) passes through a known point and the slope of its tangent at any point is given, integrate the slope to get y = ∫ (slope) dx, then substitute the point to find C.
Example dy/dx = 2x − 3 and the curve passes through (2, 5). y = x² − 3x + C; 5 = 4 − 6 + C → C = 7, so y = x² − 3x + 7.
The slope of the tangent is dy/dx; if instead the normal is given, use dy/dx = −1/(normal slope).

Comparison aspect	Main idea	Keywords
•	Acceleration a = dv/dt so v = ∫ a dt velocity v = ds/dt so s = ∫ v dt — each integration needs one initial condition (initial velocity initial position)	Accelerationvelocityintegrationneeds
Example	a body starts from rest with a = 6t − 4. v = 3t² − 4t s = t³ − 2t² at t = 3 v = 15 m/s and s = 9 m	bodystartsrest
Work done by a variable force	W = ∫ F ds and the change of momentum is ∫ F dt (see the vector-functions lesson)	changemomentumvectorfunctions

A curve has second derivative d²y/dx² = 12x − 2 and passes through (0, 1) with tangent slope 4 there. dy/dx = 6x² − 2x + C₁ → C₁ = 4. y = 2x³ − x² + 4x + C₂ → C₂ = 1. y = 2x³ − x² + 4x + 1.

Indefinite Integration of Trigonometric, Exponential and Logarithmic Functions · 6 / 11

Differentials

The differential of a function

For y = f(x) the differential is dy = f′(x) dx the differential of the function equals its derivative multiplied by the differential of the independent variable.
dx is the increment of x; dy is the corresponding change along the tangent, not along the curve. For small increments dy ≈ Δy, and the error shrinks as Δx → 0.
Second differential d²y = f″(x) dx² (only when x is the independent variable, so that dx is constant).
Rewriting a differential equation in differential form is what makes direct integration legal: dy/dx = g(x)h(y) → dy/h(y) = g(x) dx, then integrate both sides.

y = x³ ln x → dy = (3x² ln x + x²) dx.
y = sin(2x + 1) → dy = 2 cos(2x + 1) dx.
a cube of side 5 cm grows by 0.1 cm. V = x³, dV = 3x² dx = 3(25)(0.1) = 7.5 cm³ (the true increase is 7.31 cm³).
dy/dx = x/y with y(0) = 3 → y dy = x dx → y²/2 = x²/2 + C → C = 9/2, so y² = x² + 9.

Indefinite Integration of Trigonometric, Exponential and Logarithmic Functions · 7 / 11

Integration by substitution

The idea

Comparison aspect	Main idea	Keywords
It reverses the chain rule	if the integrand contains a function of g(x) together with a multiple of g′(x) put u = g(x) and du = g′(x) dx	integrandcontainsfunctiontogether
Steps	① choose u (usually the inner function the denominator or the expression under a root) ② compute du and solve for dx ③ rewrite the whole integral in u ④ integrate ⑤ return to x	chooseusuallyinnerfunction
•	For a definite integral either convert the limits to u or substitute back before applying the original limits	definiteintegraleitherconvert

∫ f′(x)/f(x) dx = ln|f(x)| + C.
∫ [f(x)]ⁿ f′(x) dx = [f(x)]ⁿ⁺¹/(n + 1) + C, n ≠ −1.

∫ 2x(x² + 1)⁵ dxu = x² + 1, du = 2x dx → ∫ u⁵ du = u⁶/6 + C = (x² + 1)⁶/6 + C.

∫ cos⁴x sin x dxu = cos x, du = −sin x dx → −∫ u⁴ du = −cos⁵x/5 + C.

∫ dx/(x ln x)u = ln x, du = dx/x → ∫ du/u = ln|ln x| + C.

∫ √(2x + 3) dxu = 2x + 3, du = 2 dx → ½·(2/3)u3/2 = (2x + 3)3/2/3 + C.

Indefinite Integration of Trigonometric, Exponential and Logarithmic Functions · 8 / 11

Integration by parts

The rule

∫ u dv = uv − ∫ v du, taken from the product rule d(uv) = u dv + v du.
Choose u by LIATE Logarithmic, Inverse trig, Algebraic (powers of x), Trigonometric, Exponential — the first type in this list becomes u; everything left, including dx, becomes dv.
Use it when the integrand is a product of two unlike factors, e.g. x e^x, x sin x, x ln x, or a single logarithm or inverse trig function.

Comparison aspect	Main idea	Keywords
∫ x ex dx	u = x dv = exdx → x ex − ∫ ex dx = x ex − ex + C	exdx
∫ x cos x dx	u = x dv = cos x dx → x sin x − ∫ sin x dx = x sin x + cos x + C
∫ ln x dx	u = ln x dv = dx → x ln x − ∫ x·(1/x) dx = x ln x − x + C
∫ x² ex dx needs parts twice	x²ex − 2x ex + 2ex + C	x²ex
•	Choosing the wrong u (for instance u = e^x in (1)) makes the new integral harder — always test whether ∫ v du is simpler than the original.	Choosingwronginstancemakes

Shape of integrand	Method	Example
Sum of powers of x	Power rule, term by term	∫ (3x² + 4x + 5) dx
Composite function × derivative of the inner	Substitution	∫ 2x(x² + 1)⁵ dx
Derivative over the function itself	Logarithm	∫ (2x + 1)/(x² + x + 4) dx
Product of two unlike factors	Parts (LIATE)	∫ x ln x dx
sin²x or cos²x	Double-angle identity first	∫ sin²x dx

Indefinite Integration of Trigonometric, Exponential and Logarithmic Functions · 9 / 11

Test yourself

∫ (4x³ − 2x + 7) dx = x⁴ − x² + 7x + C
∫ sin(3x + 1) dx = −cos(3x + 1)/3 + C
∫ sec²(4x) dx = tan(4x)/4 + C
∫ e^−x/2 dx = −2e^−x/2 + C
∫ dx/(5 − 2x) = −ln|5 − 2x|/2 + C
∫ cosec x cot x dx = −cosec x + C
∫ cot x dx = ln|sin x| + C (compare with ∫ tan x dx = ln|sec x| + C)
∫ x e^2x dx = (x e^2x)/2 − e^2x/4 + C
y = x³ − 2x + 1
(2x sin x + x² cos x) dx

Indefinite Integration of Trigonometric, Exponential and Logarithmic Functions · 10 / 11

Exam traps and key takeaways

💡 High-yield traps

∫ sin x dx = −cos x + C; the sign is the most-marked error in this lesson. ∫ cos x dx = +sin x.
Never write ∫ dx/x as ln x — it is ln|x|, and the modulus is a marked step.
∫ dx/x² is not a logarithm it is −1/x + C. The log rule only applies when the power of x is exactly one.
For a^x you must divide by ln a; forgetting it is the classic mistake on 2^x and 3^x.
Divide by the coefficient of x after integrating cos(ax + b) or e^{ax + b}; forgetting the factor a or 1/a is worth half the mark.
Always append + C to an indefinite integral; a definite integral has no constant at all.
Expand the product before integrating (e.g. (x − 1)(x + 3)); substitution is only worth the effort when a matching derivative appears.
In parts, if the new integral ∫ v du is harder than the original you chose u wrongly — swap and retry.
A differential dy = f′(x) dx is not the same as Δy; use it for approximation only when the increment is small.

Indefinite Integration of Trigonometric, Exponential and Logarithmic Functions · 11 / 11

Curriculum

Mathematics

Unit 1

Unit 2

Unit 3

Unit 4

Unit 5

Unit 6

Unit 7

Indefinite Integration of Trigonometric, Exponential and Logarithmic Functions

Indefinite Integration of Trigonometric, Exponential and Logarithmic Functions

Overview

The antiderivative and the indefinite integral

Definition

The table of standard integrals (including trigonometric)

Integrals of exponential functions

The two rules

Integrals of logarithmic functions (the 1/x rule)

The rule

Applications of the indefinite integral

Geometric: a curve from its slope

Differentials

The differential of a function

Integration by substitution

The idea

Integration by parts

The rule

Test yourself

Exam traps and key takeaways

💡 High-yield traps

Well done! You finished this lesson