Calculus: Integration

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your NECO exam.

Integration is the reverse of differentiation. If $F’(x) = f(x)$, then $F(x)$ is an antiderivative (indefinite integral) of $f(x)$.

Indefinite Integrals: $$\int f(x), dx = F(x) + C$$ where $C$ is the constant of integration.

Standard Integrals:

Function	Integral
$x^n$ ($n \neq -1$)	$\dfrac{x^{n+1}}{n+1} + C$
$\frac{1}{x}$	$\ln
$e^x$	$e^x + C$
$e^{kx}$	$\dfrac{e^{kx}}{k} + C$
$\sin x$	$-\cos x + C$
$\cos x$	$\sin x + C$
$\sec^2 x$	$\tan x + C$
$\csc^2 x$	$-\cot x + C$

Rules:

• $\int [f(x) + g(x)], dx = \int f(x), dx + \int g(x), dx$
• $\int k \cdot f(x), dx = k \int f(x), dx$

Definite Integrals: $$\int_a^b f(x), dx = F(b) - F(a)$$ where $F$ is any antiderivative of $f$.

⚡ NECO Tip: For definite integrals, evaluate the antiderivative at both limits and subtract. The area under a curve $y = f(x)$ from $x = a$ to $x = b$ is $\int_a^b f(x), dx$. If the curve is below the x-axis, the integral gives a negative value.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for NECO Mathematics students with a few days to months.

Integration by Substitution:

Let $u = g(x)$, then $du = g’(x), dx$.

Example: $\int 2x e^{x^2}, dx$

• $u = x^2 \Rightarrow du = 2x, dx$
• $\int 2x e^{x^2}, dx = \int e^u, du = e^u + C = e^{x^2} + C$

Integration by Parts: $$\int u, dv = uv - \int v, du$$

Choose $u$ using LIATE (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential).

Example: $\int x e^x, dx$

• $u = x \Rightarrow du = dx$
• $dv = e^x dx \Rightarrow v = e^x$
• $\int x e^x, dx = x e^x - \int e^x, dx = x e^x - e^x + C = e^x(x-1) + C$

Area Under a Curve: $$\text{Area} = \int_a^b f(x), dx$$

If $f(x)$ goes below the x-axis, find where it crosses and split the integral: $$\text{Net area} = \int_a^{x_0} f(x), dx + \int_{x_0}^b f(x), dx$$

Area Between Two Curves: $$\text{Area} = \int_a^b [f(x) - g(x)], dx \quad \text{where } f(x) \geq g(x)$$

Example: Area between $y = x^2$ and $y = x$ from $x = 0$ to $x = 1$: $$\int_0^1 (x - x^2), dx = \left[\frac{x^2}{2} - \frac{x^3}{3}\right]_0^1 = \frac{1}{2} - \frac{1}{3} = \frac{1}{6} \text{ square units}$$

⚡ NECO Common Mistakes:

• Forgetting the constant $C$ in indefinite integrals
• In integration by parts, choosing $u$ and $dv$ incorrectly — LIATE helps but isn’t foolproof
• Not splitting the integral when the curve crosses the x-axis
• Mixing up the formula for area between curves vs area under a single curve

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for NECO and JAMB Mathematics preparation.

Trigonometric Integrals:

$$\int \sin^2 x, dx = \int \frac{1 - \cos 2x}{2}, dx = \frac{x}{2} - \frac{\sin 2x}{4} + C$$ $$\int \cos^2 x, dx = \int \frac{1 + \cos 2x}{2}, dx = \frac{x}{2} + \frac{\sin 2x}{4} + C$$ $$\int \tan x, dx = -\ln|\cos x| + C = \ln|\sec x| + C$$ $$\int \sec x, dx = \ln|\sec x + \tan x| + C$$

Partial Fractions:

Example: $\int \frac{3x+1}{x^2-4}, dx$

1. $x^2 - 4 = (x+2)(x-2)$
2. $\frac{3x+1}{(x+2)(x-2)} = \frac{A}{x+2} + \frac{B}{x-2}$
3. $3x+1 = A(x-2) + B(x+2)$
4. Setting $x=2$: $7 = 4B \Rightarrow B = \frac{7}{4}$. Setting $x=-2$: $-5 = -4A \Rightarrow A = \frac{5}{4}$
5. $\int \frac{5/4}{x+2}, dx + \int \frac{7/4}{x-2}, dx = \frac{5}{4}\ln|x+2| + \frac{7}{4}\ln|x-2| + C$

Integration of Rational Functions of Trigonometric Functions:

For integrals of the form $\int \frac{dx}{\sin x}$ or $\int \frac{dx}{\cos x}$, multiply numerator and denominator by the conjugate.

Reduction Formulas:

$$\int \sin^n x, dx = -\frac{\sin^{n-1} x \cos x}{n} + \frac{n-1}{n}\int \sin^{n-2} x, dx$$

Volumes of Revolution:

Volume when $y = f(x)$ is rotated $360°$ about the x-axis from $x=a$ to $x=b$: $$V = \pi \int_a^b [f(x)]^2, dx$$

Volume when $x = g(y)$ is rotated $360°$ about the y-axis from $y=c$ to $y=d$: $$V = \pi \int_c^d [g(y)]^2, dy$$

Mean Value Theorem for Integrals: If $f(x)$ is continuous on $[a,b]$, there exists $c \in (a,b)$ such that: $$\int_a^b f(x), dx = f(c)(b-a)$$

NECO/JAMB Patterns:

• NECO frequently asks: evaluate definite and indefinite integrals; use substitution and integration by parts; find areas under curves and between curves; find volumes of revolution; integrate trigonometric functions

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