Integration Formulas – Complete List with Examples and Tips to Master

Integral calculus is a core component of higher mathematics and one of the most powerful tools in the analytical toolkit of students, engineers, and scientists. At the heart of integral calculus lie integration formulas, which allow us to find the area under curves, solve differential equations, and understand accumulation processes.

Whether you’re preparing for competitive exams, tackling university-level calculus, or applying mathematics in the real world, mastering integration formulas is absolutely essential. In this article, we’ll dive deep into the most important integration formulas, covering basic rules, trigonometric identities, exponential functions, logarithms, and special functions — all explained in a way that’s easy to follow and remember.

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What is Integration?

Integration is the reverse process of differentiation. While differentiation deals with rates and slopes, integration helps determine the total accumulation — such as area under a curve, displacement from velocity, or work done by a force.

In its most basic form, an integral is represented as:

$$\int f(x)dx$$

This expression reads as “the integral of $f(x)$ with respect to $x$“.

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Why Are Integration Formulas Important?

Integration formulas help simplify complex problems across:

• Engineering (signal processing, fluid flow, structural mechanics)

• Physics (motion, energy, electromagnetism)

• Economics (cost and revenue functions)

• Biology (population growth models)

• Statistics (probability distributions)

Mastering integration formulas not only saves time during exams but also equips learners with tools to handle real-life quantitative challenges.

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1. Basic Integration Formulas

These are foundational rules that every calculus student must know:

Keyword tip: These basic integration formulas form the base for more complex integrals involving trigonometric and exponential functions.

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2. Integration Formulas for Trigonometric Functions

Trigonometric integrals are common in physics and engineering problems:

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3. Exponential and Logarithmic Integration Formulas

These functions are crucial in modelling growth, decay, and compound interest:

Use these integration formulas in problems involving radioactive decay, population dynamics, and economic models.

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4. Special Integration Formulas

These integrals often arise in advanced calculus problems:

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5. Integration by Parts Formula

Integration by parts is a key formula based on the product rule of differentiation:

$$\int udv=uv-\int vdu$$

Steps:

1. Identify $u$ and $dv$

2. Differentiate $u$, integrate $dv$

3. Plug into the formula

Example:

∫xexdx=xex−∫exdx=xex−ex+C\int x e^x dx = x e^x – \int e^x dx = x e^x – e^x + C∫xexdx=xex−∫exdx=xex−ex+C

Apply this formula when dealing with product of functions, like $x\sin x$, $x\ln x$, or xexx e^xxex.

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6. Integration by Substitution

Use substitution when integrals involve a composite function:

∫f(g(x))g′(x)dx=∫f(u)du\int f(g(x))g'(x) dx = \int f(u) du∫f(g(x))g′(x)dx=∫f(u)du

Example:

∫2xcos⁡(x2)dxLet u=x2⇒du=2xdx\int 2x \cos(x^2) dx \quad \text{Let } u = x^2 \Rightarrow du = 2x dx∫2xcos(x2)dxLet u=x2⇒du=2xdx =∫cos⁡(u)du=sin⁡(u)+C=sin⁡(x2)+C= \int \cos(u) du = \sin(u) + C = \sin(x^2) + C=∫cos(u)du=sin(u)+C=sin(x2)+C

This method simplifies seemingly complex integration formulas into standard forms.

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7. Definite Integration Formulas

Definite integrals compute exact values over an interval:

∫abf(x)dx\int_a^b f(x) dx∫ab​f(x)dx

Properties:

• ∫aaf(x)dx=0\int_a^a f(x) dx = 0∫aa​f(x)dx=0

• ∫abf(x)dx=−∫baf(x)dx\int_a^b f(x) dx = -\int_b^a f(x) dx∫ab​f(x)dx=−∫ba​f(x)dx

• ∫ab[f(x)+g(x)]dx=∫abf(x)dx+∫abg(x)dx\int_a^b [f(x) + g(x)] dx = \int_a^b f(x) dx + \int_a^b g(x) dx∫ab​[f(x)+g(x)]dx=∫ab​f(x)dx+∫ab​g(x)dx

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8. Area Under Curves Using Integration

One of the most common applications:

Area=∫abf(x)dx\text{Area} = \int_a^b f(x) dxArea=∫ab​f(x)dx

Example:
Find the area under y=x2y = x^2y=x2 from 0 to 2:

∫02x2dx=[x33]02=83\int_0^2 x^2 dx = \left[\frac{x^3}{3}\right]_0^2 = \frac{8}{3}∫02​x2dx=[3x3​]02​=38​

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9. Integration Formulas for Rational Functions

These often use partial fractions:

∫1(x+a)(x+b)dx\int \frac{1}{(x+a)(x+b)} dx∫(x+a)(x+b)1​dx

Steps:

1. Split into partial fractions

2. Integrate each term separately

Example:

∫1x(x+1)dx=∫(1x−1x+1)dx=ln⁡∣x∣−ln⁡∣x+1∣+C\int \frac{1}{x(x+1)} dx = \int \left(\frac{1}{x} – \frac{1}{x+1}\right) dx = \ln|x| – \ln|x+1| + C∫x(x+1)1​dx=∫(x1​−x+11​)dx=ln∣x∣−ln∣x+1∣+C

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10. Tricky Substitutions and Tricks

Some tough integrals need clever manipulation:

• ∫a2−x2dx\int \sqrt{a^2 – x^2} dx∫a2−x2​dx → use $x=a\sin \theta$

• ∫x2+a2dx\int \sqrt{x^2 + a^2} dx∫x2+a2​dx → use $x=a\tan \theta$

• ∫x2−a2dx\int \sqrt{x^2 – a^2} dx∫x2−a2​dx → use $x=a\sec \theta$

These often appear in engineering problems involving arc lengths and geometry.

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Real-World Applications of Integration Formulas

1. Physics: Work, motion under gravity, centre of mass

2. Economics: Calculating total cost and consumer surplus

3. Engineering: Fluid flow, stress analysis, thermodynamics

4. Biology: Growth models, heart rate analysis, drug concentration

5. Computer Graphics: Surface modeling and shading techniques

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Tips to Memorise Integration Formulas

• Create visual formula charts or flashcards

• Group formulas by category (e.g., trig, exponential)

• Practice 5–10 problems daily using different methods

• Use memory aids or mnemonics

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Final Thoughts

Mastering integration formulas is not just about memorising equations — it’s about recognising patterns, applying techniques, and gaining confidence in tackling real problems. This guide serves as a comprehensive yet simple reference point for all your integral calculus needs.

Whether you’re a high school student preparing for boards or a college student diving into advanced calculus, this SEO-rich article on integration formulas offers clarity, context, and actionable insights.