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Indefinite Integrals

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Indefinite Integrals

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Indefinite Integrals

🎯 Learning Objectives

By the end of this topic, you will be able to:

Define an anti-derivative and the indefinite integral
Apply basic integration rules (power, exponential, trigonometric)
Find indefinite integrals using substitution method
Recognize the constant of integration $C$

📋 Prerequisites

Derivatives — differentiation rules (integration is the reverse)

📖 Core Content

15.1 Intuition: The Reverse of Differentiation

If differentiation tells you the rate of change, integration tells you the original quantity given its rate. If velocity

v(t)

is the derivative of position

s(t)

, then position is the integral of velocity. An anti-derivative of

f(x)

is a function

F(x)

such that

F'(x) = f(x)

. The indefinite integral is the set of all anti-derivatives:

\int f(x)\,dx = F(x) + C

where

C

is the constant of integration (since derivative of a constant is

0

💡 Why this matters: Integration is used everywhere: computing areas, volumes, work, probability (cumulative distribution functions), and solving differential equations.

15.2 Basic Integration Rules

Derivative Rule	Integral Rule
$\frac{d}{dx}C = 0$	$\int 0\,dx = C$
$\frac{d}{dx}x^n = nx^{n-1}$	$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\ (n \neq -1)$
$\frac{d}{dx}\ln	x
$\frac{d}{dx}e^x = e^x$	$\int e^x\,dx = e^x + C$
$\frac{d}{dx}a^x = a^x\ln a$	$\int a^x\,dx = \frac{a^x}{\ln a} + C$
$\frac{d}{dx}\sin x = \cos x$	$\int \cos x\,dx = \sin x + C$
$\frac{d}{dx}\cos x = -\sin x$	$\int \sin x\,dx = -\cos x + C$
$\frac{d}{dx}\tan x = \sec^2 x$	$\int \sec^2 x\,dx = \tan x + C$

Linearity:

\int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx

\int cf(x)\,dx = c\int f(x)\,dx

15.3 Worked Examples

Example 1.1: Find

\int (3x^2 - 2x + 1)\,dx

. \int (3x^2 - 2x + 1)\,dx &= 3\int x^2\,dx - 2\int x\,dx + \int 1\,dx \\ &= 3\cdot\frac{x^3}{3} - 2\cdot\frac{x^2}{2} + x + C \\ &= x^3 - x^2 + x + C \end{aligned}

\\boxed{x^3 - x^2 + x + C} **Example 1.2:** Find $\\int \\left(e^x + \\frac{1}{x}\\right)\\,dx$. \\int e^x,dx + \\int\\frac{1}{x},dx = e^x + \\ln|x| + C \\boxed{e^x + \\ln|x| + C} **Example 1.3:** Find $\\int \\sqrt\[3\]{x^2}\\,dx$. \\int x^{2/3},dx = \\frac{x^{5/3}}{5/3} + C = \\frac{3}{5}x^{5/3} + C \\boxed{\\frac{3}{5}x^{5/3} + C} \--- ### 15.4 Substitution (Reverse Chain Rule) When the integrand has the form $f'(g(x))\\cdot g'(x)$, substitute $u = g(x)$: \\int f'(g(x)) \\cdot g'(x),dx = \\int f'(u),du = f(u) + C = f(g(x)) + C **Recipe:** 1. Choose $u = g(x)$ (the "inner function") 2. Compute $du = g'(x)\\,dx$ 3. Rewrite integral in terms of $u$ 4. Integrate 5. Substitute back **Example 2.1:** Find $\\int 2x\\,e^{x^2}\\,dx$. Let $u = x^2$, $du = 2x\\,dx$. $\\int e^u\\,du = e^u + C = e^{x^2} + C$ \\boxed{e^{x^2} + C} **Example 2.2:** Find $\\int \\frac{2x}{x^2+1}\\,dx$. Let $u = x^2+1$, $du = 2x\\,dx$. $\\int \\frac{1}{u}\\,du = \\ln|u| + C = \\ln|x^2+1| + C$ \\boxed{\\ln(x^2+1) + C} **Example 2.3:** Find $\\int \\cos(3x)\\,dx$. Let $u = 3x$, $du = 3\\,dx \\implies dx = du/3$. $\\int \\cos u \\cdot \\frac{du}{3} = \\frac{1}{3}\\sin u + C = \\frac{1}{3}\\sin(3x) + C$ \\boxed{\\frac{1}{3}\\sin(3x) + C} \--- ## 📐 Key Formulas — Summary Table | Integral | Result | Condition | |:---|:---|:---| | $\\int x^n\\,dx$ | $\\frac{x^{n+1}}{n+1} + C$ | $n \\neq -1$ | | $\\int \\frac{1}{x}\\,dx$ | $\\ln|x| + C$ | | | $\\int e^x\\,dx$ | $e^x + C$ | | | $\\int a^x\\,dx$ | $\\frac{a^x}{\\ln a} + C$ | $a > 0, a \\neq 1$ | | $\\int \\sin x\\,dx$ | $-\\cos x + C$ | | | $\\int \\cos x\\,dx$ | $\\sin x + C$ | | | $\\int \\sec^2 x\\,dx$ | $\\tan x + C$ | | | Substitution | $\\int f(g(x))g'(x)\\,dx = \\int f(u)\\,du$ | $u = g(x)$ | --- ## ⚠️ Common Pitfalls ### Pitfall 1: Forgetting the Constant $C$ Every indefinite integral must include $+C$. Differentiating any constant gives $0$, so the integral is only determined up to a constant. ### Pitfall 2: Misapplying Power Rule for $n = -1$ $\\int x^{-1}\\,dx = \\ln|x| + C$, NOT $\\frac{x^0}{0}$ (which is undefined). ### Pitfall 3: Forgetting to Divide by the Coefficient in Substitution When substituting $u = ax$, $dx = du/a$. **The $1/a$ factor is essential.** --- ## 📝 Practice Questions > **Q1: Find $\\int x^4\\,dx$.** > > $\\frac{x^5}{5} + C$ > >

\boxed{\frac{x^5}{5} + C}

> **Q2: Find $\\int (4x^3 - 2x + 7)\\,dx$.** > > $x^4 - x^2 + 7x + C$ > >

\boxed{x^4 - x^2 + 7x + C}

> **Q3: Find $\\int \\frac{1}{x^2}\\,dx$.** > > $\\int x^{-2}\\,dx = \\frac{x^{-1}}{-1} + C = -\\frac{1}{x} + C$ > >

\boxed{-\frac{1}{x} + C}

> **Q4: Find $\\int e^{2x}\\,dx$.** > > Let $u = 2x$, $du = 2\\,dx$, $dx = du/2$. > $\\frac{1}{2}\\int e^u\\,du = \\frac{1}{2}e^{2x} + C$ > >

\boxed{\frac{1}{2}e^{2x} + C}

> **Q5: Find $\\int 3\\sin x\\,dx$.** > > $-3\\cos x + C$ > >

\boxed{-3\cos x + C}

> **Q6: Find $\\int \\frac{2x}{x^2+1}\\,dx$.** > > Let $u = x^2+1$, $du = 2x\\,dx$. $\\int \\frac{du}{u} = \\ln|u| + C = \\ln(x^2+1) + C$ > >

\boxed{\ln(x^2+1) + C}

> **Q7: Find $\\int \\sqrt{x}\\,dx$.** > > $\\int x^{1/2}\\,dx = \\frac{x^{3/2}}{3/2} + C = \\frac{2}{3}x^{3/2} + C$ > >

\boxed{\frac{2}{3}x^{3/2} + C}

> **Q8: Find $\\int (x+1)^4\\,dx$.** > > Let $u = x+1$, $du = dx$. $\\int u^4\\,du = \\frac{u^5}{5} + C = \\frac{(x+1)^5}{5} + C$ > >

\boxed{\frac{(x+1)^5}{5} + C}

> **Q9: Find $\\int \\frac{1}{3x+2}\\,dx$.** > > Let $u = 3x+2$, $du = 3\\,dx$, $dx = du/3$. > $\\frac{1}{3}\\int \\frac{du}{u} = \\frac{1}{3}\\ln|3x+2| + C$ > >

\boxed{\frac{1}{3}\ln|3x+2| + C}

> **Q10: Find $\\int \\sin(2x)\\cos x\\,dx$.** > > This requires trig identity: $\\sin(2x)\\cos x = 2\\sin x\\cos^2 x$, then substitution $u = \\cos x$. > Let $u = \\cos x$, $du = -\\sin x\\,dx$. > $-2\\int u^2\\,du = -2\\cdot\\frac{u^3}{3} + C = -\\frac{2}{3}\\cos^3 x + C$ > >

\boxed{-\frac{2}{3}\cos^3 x + C}$$ --- ## 🔗 Cross-References - Next topic: [Definite Integrals & FTC](integration-definite.md) - Related: [Integration Techniques](integration-techniques.md) - Across courses: BSMA1003 Maths 2 (multivariable integration); BSMA1002 Stats 1 (probability densities) Join Discord Previous8.2 Applications of DerivativesNext9.2 Definite Integrals & FTC