“Calculus: Integration”

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Calculus: Integration — Quick Facts for JAMB

Indefinite Integrals: Integration is the reverse of differentiation.

• $\int x^n,dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$)
• $\int \frac{1}{x},dx = \ln|x| + C$
• $\int e^x,dx = e^x + C$
• $\int \sin x,dx = -\cos x + C$
• $\int \cos x,dx = \sin x + C$

Definite Integrals: $\int_a^b f(x),dx = [F(x)]_a^b = F(b) - F(a)$ where $F’(x) = f(x)$. The constant $C$ cancels out in definite integrals.

Basic Rules:

• $\int [f(x) + g(x)],dx = \int f(x),dx + \int g(x),dx$
• $\int k f(x),dx = k \int f(x),dx$ (constant multiple rule)
• $\int_a^b f(x),dx = -\int_b^a f(x),dx$ (reversing limits changes sign)

⚡ Exam tip: Always include the constant of integration $C$ for indefinite integrals. Don’t forget it!

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Calculus: Integration — JAMB UTME Study Guide

Integration by Substitution: For $\int f(g(x)) \cdot g’(x),dx$, let $u = g(x)$, then $\int f(u),du$.

Example: $\int 2x \cos(x^2),dx$. Let $u = x^2$, so $du = 2x,dx$. $\int \cos(u),du = \sin(u) + C = \sin(x^2) + C$.

Integration by Parts: $\int u,dv = uv - \int v,du$. Choose $u$ using LIATE: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential.

Example: $\int x e^x,dx$. Let $u = x$ (algebraic), $dv = e^x dx$ (exponential). Then $du = dx$, $v = e^x$. $\int x e^x,dx = x e^x - \int e^x,dx = x e^x - e^x + C = e^x(x-1) + C$.

Trigonometric Integrals:

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

Partial Fractions Before Integrating: Example: $\int \frac{1}{x^2-1},dx$. Partial fractions: $\frac{1}{x^2-1} = \frac{1/2}{x-1} - \frac{1/2}{x+1}$. $\int = \frac{1}{2}\ln|x-1| - \frac{1}{2}\ln|x+1| + C = \frac{1}{2}\ln\left|\frac{x-1}{x+1}\right| + C$.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Calculus: Integration — Comprehensive Mathematics Notes

Definite Integrals — Area Under a Curve:

The definite integral $\int_a^b f(x),dx$ represents the net signed area between $y = f(x)$ and the x-axis from $x = a$ to $x = b$. Area above the x-axis contributes positively; area below contributes negatively.

For area between two curves: $\int_a^b [f(x) - g(x)],dx$ where $f(x) \geq g(x)$ on $[a,b]$.

Example: Find area bounded by $y = x^2$ and $y = 4$. Intersection: $x^2 = 4$, so $x = -2$ and $x = 2$. Area = $\int_{-2}^{2} (4 - x^2),dx = [4x - x^3/3]_{-2}^{2} = (8 - 8/3) - (-8 + 8/3) = (16/3) - (-16/3) = 32/3$ square units.

Mean Value Theorem for Integrals: If $f(x)$ is continuous on $[a,b]$, then there exists $c \in [a,b]$ such that $\int_a^b f(x),dx = f(c)(b-a)$. The average value of $f$ on $[a,b]$ is $\frac{1}{b-a}\int_a^b f(x),dx$.

Volumes of Revolution:

• About x-axis: $V = \pi \int_a^b [f(x)]^2,dx$
• About y-axis: $V = 2\pi \int_a^b x f(x),dx$

Example: $y = \sqrt{x}$ from $x = 0$ to $x = 4$ rotated about x-axis. $V = \pi \int_0^4 (\sqrt{x})^2,dx = \pi \int_0^4 x,dx = \pi [x^2/2]_0^4 = \pi(8) = 8\pi$ cubic units.

Integration by Parts — Tabular Method (Repeated Integration by Parts):

For $\int x^2 e^x,dx$:

Sign	D (differentiate)	I (integrate)
$+$	$x^2$	$e^x$
$-$	$2x$	$e^x$
$+$	$2$	$e^x$
$-$	$0$	$e^x$

Result: $x^2 e^x - 2x e^x + 2e^x + C = e^x(x^2 - 2x + 2) + C$.

Reduction Formulas: For $I_n = \int \sin^n x,dx$: $I_n = -\frac{\sin^{n-1}x \cos x}{n} + \frac{n-1}{n} I_{n-2}$. This reduces the power until you can integrate.

Differential Equations — Separable: $\frac{dy}{dx} = f(x)g(y)$. Separate: $\frac{dy}{g(y)} = f(x),dx$. Integrate both sides.

Example: $\frac{dy}{dx} = xy$. $\frac{dy}{y} = x,dx$. $\ln|y| = x^2/2 + C$. $y = Ce^{x^2/2}$.

Trapezium Rule (Approximate Integration): For $n$ strips (even number for accuracy): $\int_a^b f(x),dx \approx \frac{h}{2}[y_0 + 2(y_1+y_2+…+y_{n-1}) + y_n]$ where $h = (b-a)/n$.

Example: $\int_0}^{2} x^2,dx$ with 4 strips. $h = 0.5$. $y_0 = 0$, $y_1 = 0.25$, $y_2 = 1$, $y_3 = 2.25$, $y_4 = 4$. Approximation: $\frac{0.5}{2}[0 + 2(0.25+1+2.25) + 4] = 0.25[0 + 2(3.5) + 4] = 0.25[7+4] = 0.25[11] = 2.75$. Exact: $[x^3/3]_0^2 = 8/3 = 2.667$. Error = 0.083.

JAMB Pattern Analysis: JAMB questions frequently test: (1) Basic indefinite integrals, (2) Definite integrals with $F(b)-F(a)$, (3) Integration by substitution, (4) Integration by parts for products of polynomials and exponentials/trig, (5) Finding areas under curves. Common mistake: forgetting the constant $C$ or applying the wrong sign in trigonometric integrals. JAMB 2023: “Evaluate $\int_0^1 (x^2 + 2x),dx$.” Answer: $[x^3/3 + x^2]_0^1 = (1/3 + 1) - 0 = 4/3$.

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📊 JAMB Exam Essentials

Detail	Value
Questions	180 MCQs (UTME)
Subjects	4 subjects (language + 3 for course)
Time	2 hours
Marking	+1 per correct answer
Score	400 max (used for university admission)
Registration	January – February each year

🎯 High-Yield Topics for JAMB

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📝 Previous Year Question Patterns

• Q: “The process of photosynthesis requires…” [2024 Biology]
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• Q: “Find the value of x if 2x + 5 = 15…” [2024 Mathematics]

💡 Pro Tips

• Use of English carries the most weight — master grammar rules and comprehension strategies
• JAMB syllabus is your Bible — questions come directly from it. Download and use it.
• Past questions are highly predictive — repeat patterns appear every year
• For Science students, Biology and Chemistry are high-scoring if you study NCERT-level content

🔗 Official Resources

• JAMB Official
• JAMB Syllabus

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