Calculus: Integration

🟢 Lite — Quick Review (1h–1d)

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

Integration is the reverse process of differentiation. If you differentiate $f(x)$ and get $f’(x)$, then integrating $f’(x)$ brings you back to $f(x)$, up to a constant $C$. The indefinite integral of $f(x)$ is written as $\int f(x), dx$.

Core Formulas to Memorise

Function	Integral
$\int x^n, dx$	$\dfrac{x^{n+1}}{n+1} + C$ (where $n \neq -1$)
$\int \frac{1}{x}, dx$	$\ln
$\int e^x, dx$	$e^x + C$
$\int \sin x, dx$	$-\cos x + C$
$\int \cos x, dx$	$\sin x + C$
$\int e^{kx}, dx$	$\dfrac{e^{kx}}{k} + C$
$\int a^x, dx$	$\dfrac{a^x}{\ln a} + C$

Key Rules

• Constant multiple: $\int k \cdot f(x), dx = k \int f(x), dx$
• Sum/difference: $\int [f(x) \pm g(x)], dx = \int f(x), dx \pm \int g(x), dx$

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

⚡ NABTEB Exam Tip: When you see “evaluate” or “find the area under the curve between $x=a$ and $x=b$”, they’re asking for a definite integral. Always substitute the limits correctly — upper limit first, then subtract lower limit. Watch out for negative areas: if the curve dips below the x-axis, the integral gives a negative value.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding before the NABTEB examination.

The Fundamental Theorem of Calculus

The two parts connect differentiation and integration:

1. If $F(x) = \int_a^x f(t), dt$, then $F’(x) = f(x)$
2. $\int_a^b f(x), dx = F(b) - F(a)$, where $F$ is any antiderivative of $f$

Integration by Substitution

When an integral looks complex, try substitution. The idea: let $u = g(x)$, then $du = g’(x), dx$.

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

• Let $u = x^2$, so $du = 2x, dx$
• The integral becomes $\int e^u, du = e^u + C = e^{x^2} + C$

Integration by Parts

For products of functions, use: $\int u, dv = uv - \int v, du$

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

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

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

Applications of Integration

• Area under a curve: $\int_a^b y, dx$ gives the net area between $y = f(x)$ and the x-axis from $x=a$ to $x=b$
• Area between two curves: $\int_a^b [f(x) - g(x)], dx$ (where $f(x) \geq g(x)$)
• Volume of revolution: $V = \pi \int_a^b [f(x)]^2, dx$ rotated about the x-axis

NABTEB Common Mistakes:

• Forgetting the constant of integration $C$ in indefinite integrals
• Mixing up limits in definite integrals
• For integration by parts, choosing the wrong $u$ and $dv$ — LIATE helps
• Not checking if the region is entirely above the x-axis before interpreting a negative integral as an area

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory for thorough NABTEB preparation.

Derivation of Basic Integration Formulas

Integration is the inverse of differentiation. Starting from the power rule for differentiation:

If $\dfrac{d}{dx}\left(\dfrac{x^{n+1}}{n+1}\right) = x^n$ (for $n \neq -1$), then:

$$\int x^n, dx = \frac{x^{n+1}}{n+1} + C$$

For the case $n = -1$: $\dfrac{d}{dx}(\ln x) = \dfrac{1}{x}$, so $\int \dfrac{1}{x}, dx = \ln|x| + C$.

Integration of Trigonometric Functions

$$\int \sin x, dx = -\cos x + C, \quad \int \cos x, dx = \sin x + C$$ $$\int \sec^2 x, dx = \tan x + C, \quad \int \csc^2 x, dx = -\cot x + C$$ $$\int \tan x, dx = -\ln|\cos x| + C, \quad \int \cot x, dx = \ln|\sin x| + C$$

Integration of $\sec x$ and $\csc x$:

$$\int \sec x, dx = \ln|\sec x + \tan x| + C$$ $$\int \csc x, dx = \ln|\csc x - \cot x| + C$$

Partial Fractions

For rational functions where the denominator can be factorised:

Example: $\int \dfrac{2x+1}{x^2 - 5x + 6}, dx$

1. Factor denominator: $x^2 - 5x + 6 = (x-2)(x-3)$
2. Write: $\dfrac{2x+1}{(x-2)(x-3)} = \dfrac{A}{x-2} + \dfrac{B}{x-3}$
3. Solve: $2x+1 = A(x-3) + B(x-2)$; setting $x=3$: $7=B$; setting $x=2$: $5=A(-1) \Rightarrow A=-5$
4. Integral: $-5\int \frac{dx}{x-2} + 7\int \frac{dx}{x-3} = -5\ln|x-2| + 7\ln|x-3| + C$

Reduction Formulas

For powers of trigonometric functions: $$\int \sin^n x, dx = -\frac{\sin^{n-1} x \cos x}{n} + \frac{n-1}{n}\int \sin^{n-2} x, dx$$

Numerical Integration (Trapezium Rule)

When an integral cannot be solved analytically: $$\int_a^b f(x), dx \approx \frac{h}{2}\left[f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)\right]$$ where $h = \frac{b-a}{n}$ and $x_i = a + ih$.

NABTEB Patterns:

• Questions on definite integrals and areas under curves appear frequently in Section B
• Always verify antiderivatives by differentiating your answer
• Integration by substitution typically appears as a two-step process

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