From Fundamentals to Advanced: 50+ Integration Formulas Explored

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Introduction

Unlock the power of mathematical analysis with Integration Formulas, a crucial element in calculus. Integration is a mathematical operation that involves finding the integral of a function, representing the area under its curve on a given interval. It is the reverse process of differentiation and plays a pivotal role in calculus. Understanding integration is essential for solving problems related to accumulation, motion, and change. From calculating areas to predicting future values, integration is a versatile tool. Let’s delve into the fundamental aspects of integration and explore a comprehensive list of basic integration formulas.

List of Basic Integration Formula

Since \( \frac{d}{dx}(F(x))=f(x) \iff \int f(x)\,\mathrm{d}x = F(x) +c \), based upon this definition and various standard differentiation formulas, we obtain the following integration formulae:

\begin{align*} 1. & \quad \int x^n \,\mathrm{d}x = \frac{x^{n+1}}{n+1}+C \quad (n \neq -1) \\[0.2cm] 2. & \quad \int \frac{1}{x} \,\mathrm{d}x = \log x+C \\[0.2cm]3. & \quad \int e^x \,\mathrm{d}x = e^x+C \\[0.2cm] 4. & \quad \int a^x \,\mathrm{d}x = \frac{a^x}{\log a}+C \\[0.2cm] 5. & \quad \int \sin x \,\mathrm{d}x = -\cos x+C \\[0.2cm] 6. & \quad \int \cos x \,\mathrm{d}x = \sin x+C \\[0.2cm] 7. & \quad \int \sec^2 x \,\mathrm{d}x = \tan x +C\\[0.2cm] 8. & \quad \int \csc^2 x \,\mathrm{d}x = -\cot x +C \\[0.2cm] 9. & \quad \int \sec x \tan x \,\mathrm{d}x = \sec x+C \\[0.2cm] 10. & \quad \int \csc x \cot x \,\mathrm{d}x = -\csc x+C \\[0.2cm] 11. & \quad \int \frac{1}{\sqrt{1-x^2}} \,\mathrm{d}x = \sin^{-1} x +C \\[0.2cm] 12. & \quad \int \frac{1}{1+x^2} \,\mathrm{d}x = \tan^{-1} x +C \\[0.2cm] 13. & \quad \int \frac{1}{x \sqrt{x^2-1}} \,\mathrm{d}x = \sec^{-1} x +C \end{align*}

Basic Properties of Indefinite Integration

\begin{align*} 1. & \quad \int kf(x) \,\mathrm{d}x = k\int f(x) \, \mathrm{d}x, \text{ where k is a constant.}\\[0.2cm] 2. & \quad \int \left( f_1(x)\pm f_2(x)\pm \cdots \pm f_n(x) \right) \,\mathrm{d}x = \int f_1(x) \,\mathrm{d}x \pm \int f_2(x) \,\mathrm{d}x \pm \cdots \pm \int f_n(x) \,\mathrm{d}x\end{align*}

Integration of Function \( f(ax+b) \)

\begin{align*}\int f(x) \, \mathrm{d}x =F(x) +C \Longrightarrow \int f(ax+b)\, \mathrm{d}x=\frac{F(ax+b)}{a}+C \end{align*}

Hence the following results.

\begin{align*}1.& \quad \int \sin(ax+b)\,\mathrm{d}x=-\frac 1a \cos(ax+b)+C\\[0.2cm]2.& \quad \int \cos(ax+b)\,\mathrm{d}x=\frac 1a \sin(ax+b)+C\\[0.2cm]3.& \quad \int e^{ax}\,\mathrm{d}x=\frac 1ae^{ax}+C\\[0.2cm] 4.& \quad \int \frac{1}{ax+b}\, \mathrm{d}x=\frac 1a\log(ax+b)+C\\[0.2cm] 5.& \quad \int (ax+b)^n\,\mathrm{d}x=\frac{(ax+b)^{n+1}}{a(n+1)}+C\end{align*}

Integration of Function involving \( f'(x) \)

\begin{align*} 1. & \quad \int \left[ f(x) \right]^n f'(x)\, \mathrm{d}x = \frac{\left[ f(x) \right]^{n+1}}{n+1}+C\\[0.2cm] 2. & \quad \int e^{f(x)}f'(x)\, \mathrm{d}x = e^{f(x)}+C\\[0.2cm] 3. & \quad \int \frac{f'(x)}{f(x)}\, \mathrm{d}x = \ln (f(x)) +C \end{align*}

Integration of \( \tan x, \cot x, \sec x \,\& \csc x \)

\begin{align*} 1. & \quad \int \tan x\, \mathrm{d}x=\log|\sec x |+C\\[0.2cm]2.& \quad \int \cot x\, \mathrm{d}x=\log|\sin x|+C\\[0.2cm]3.& \quad \int \sec x\, \mathrm{d}x=\log|\sec x + \tan x|+C=\log \left| \tan \left( \frac{\pi}{4}+\frac x2 \right) \right|+C\\[0.2cm]4.& \quad \int \csc x\, \mathrm{d}x=\log|\csc x-\cot x|+C = \log\left|\tan \left( \frac x2 \right) \right|+C\end{align*}

Integration By Parts

Integration by Parts is a technique used in calculus to evaluate the integral of a product of two functions. The formula for Integration by Parts is derived from the product rule for differentiation and is expressed as:

\[ \int u \,\mathrm{d}v =uv-\int v \,\mathrm{d}u \]

where:

\( u \) and \( \mathrm{d}v \) are differentiable functions of \( x \),
\( \mathrm{d}u \) is the derivative of \( u \) with respect to \( x \),
\( v \) is the antiderivative (or integral) of \( dv \) with respect to \( x \).

The formula is often remembered using the acronym “LIATE,” which stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential functions. When choosing \( u \) and \( dv \) in the formula, it is generally recommended to select \( u \) based on the order of functions in the LIATE mnemonic, as this helps simplify the integral.

Here are some standard formulae where the Integration by Parts rule is used:

\begin{align*} 1. & \quad \int \ln x \, \mathrm{d}x = x \ln x - x + C \\[0.2cm] 2. & \quad \int \tan^{-1} x \, \mathrm{d}x = x \tan^{-1} x - \frac{1}{2} \ln (1+x^2) + C \\[0.2cm] 3. & \quad \int \sin^{-1} x \, \mathrm{d}x = x \sin^{-1} x + \sqrt{1-x^2} + C \\[0.2cm] 4. & \quad \int \cos^{-1} x \, \mathrm{d}x = x \cos^{-1} x - \sqrt{1-x^2} + C \\[0.2cm] 5. & \quad \int e^{ax}\sin(bx+c) \, \mathrm{d}x = \frac{e^{ax}}{a^2+b^2}\left(a\sin(bx+c) - b\cos(bx+c)\right) + C \\[0.2cm] 6. & \quad \int e^{ax}\cos(bx+c) \, \mathrm{d}x = \frac{e^{ax}}{a^2+b^2}\left(a\cos(bx+c) + b\sin(bx+c)\right) + C \end{align*}

Click here to see how Integrating by Parts rule is used to find integration of \( \ln x \)

Integral of some Special Fucntions

\begin{align*} 1. & \quad \int \frac{1}{x^2 + a^2} \, \mathrm{d}x = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C \\[0.2cm] 2. & \quad \int \frac{1}{a^2 - x^2} \, \mathrm{d}x = \frac{1}{2a} \log\left|\frac{a+x}{a-x}\right| + C \\[0.2cm] 3. & \quad \int \frac{1}{x^2 - a^2} \, \mathrm{d}x = \frac{1}{2a} \log\left|\frac{x-a}{x+a}\right| + C \\[0.2cm] 4. & \quad \int \frac{1}{\sqrt{a^2 - x^2}} \, \mathrm{d}x = \sin^{-1}\left(\frac{x}{a}\right) + C \\[0.2cm] 5. & \quad \int \frac{1}{\sqrt{x^2 + a^2}} \, \mathrm{d}x = \sinh^{-1}\left(\frac{x}{a}\right) + C = \log\left(x+\sqrt{x^2+a^2}\right)+C\\[0.2cm] 6. & \quad \int \frac{1}{\sqrt{x^2 - a^2}} \, \mathrm{d}x = \cosh^{-1}\left(\frac{x}{a}\right) + C= \log\left(x+\sqrt{x^2-a^2}\right)+C \\[0.2cm] 7. & \quad \int \sqrt{a^2 - x^2} \, \mathrm{d}x = \frac{1}{2}\left(x\sqrt{a^2-x^2} + a^2\sin^{-1}\left(\frac{x}{a}\right) \right) + C\\[0.2cm] 8. & \quad \int \sqrt{x^2 + a^2} \, \mathrm{d}x = \frac{1}{2}\left(x\sqrt{x^2+a^2} + a^2\log|x+\sqrt{x^2+a^2}| \right) + C = \frac{1}{2} \left(x\sqrt{x^2 + a^2} + a^2 \sinh^{-1}\left(\frac{x}{a}\right)\right) + C\\[0.2cm] 9. & \quad \int \sqrt{x^2 - a^2} \, \mathrm{d}x = \frac{1}{2}\left(x\sqrt{x^2-a^2} - a^2\log|x+\sqrt{x^2-a^2}| \right) + C = \frac{1}{2} \left(x\sqrt{x^2 - a^2} - a^2 \cosh^{-1}\left(\frac{x}{a}\right)\right) + C \end{align*}

Reduction Formulae

\begin{align*} 1. & \quad \int \sin^n x \, \mathrm{d}x = -\frac{1}{n} \cdot \sin^{n-1} x \cdot \cos x + \frac{n-1}{n} \cdot \int \sin^{n-2} x \, \mathrm{d}x \\[0.2cm] 2. & \quad \int \cos^n x \, \mathrm{d}x = \frac{1}{n} \cdot \cos^{n-1} x \cdot \sin x + \frac{n-1}{n} \cdot \int \cos^{n-2} x \, \mathrm{d}x \\[0.2cm] 3. & \quad \int \tan^n x \, \mathrm{d}x = \frac{1}{n-1} \cdot \tan^{n-1} x - \int \tan^{n-2} x \, \mathrm{d}x \\[0.2cm] 4. & \quad \int \cot^n x \, \mathrm{d}x = -\frac{1}{n-1} \cdot \cot^{n-1} x - \int \cot^{n-2} x \, \mathrm{d}x \\[0.2cm] 5. & \quad \int \sec^n x \, \mathrm{d}x = \frac{1}{n-1} \cdot \sec^{n-2} x \cdot \tan x + \frac{n-2}{n-1} \cdot \int \sec^{n-2} x \, \mathrm{d}x \\[0.2cm] 6. & \quad \int \csc^n x \, \mathrm{d}x = -\frac{1}{n-1} \cdot \csc^{n-2} x \cdot \cot x + \frac{n-2}{n-1} \cdot \int \csc^{n-2} x \,\mathrm{d}x \\[0.2cm] 7. & \quad \int \sin^m x \cos^n x \, \mathrm{d}x = -\frac{\sin^{m-1}x \cos^{n+1}x}{m+n}+\frac{m-1}{m+n}\int \sin^{m-2}x \cos^n x \, \mathrm{d}x \\[0.2cm] 8. & \quad \int e^{a x} \sin ^n b x \,\mathrm{d}x=\frac{a \sin b x-n b \cos b x}{a^2+n^2 b^2} e^{a x} \sin ^{n-1} b x +\frac{n(n-1)}{a^2+n^2 b^2} b^2 \int e^{a x} \sin ^{n-2} b x \,\mathrm{d}x \\[0.2cm] 9. & \quad \int e^{a x} \cos ^n b x \,\mathrm{d}x=\frac{a \cos b x+n b \sin b x}{a^2+n^2 b^2} e^{a x} \cos ^{n-1} b x +\frac{n(n-1)}{a^2+n^2 b^2} b^2 \int e^{a x} \cos ^{n-2} b x \,\mathrm{d}x \\[0.2cm] 10. & \quad \int \frac{1}{\left( x^2+k \right)^n}\, \mathrm{d}x = \frac{x}{(2n-2)k\left(x^2+k\right)^{n-1}}+\frac{2n-3}{(2n-2)k}\int \frac{1}{\left( x^2+k \right)^{n-1}} \, \mathrm{d}x \end{align*}

Walli's Integral Formula

\begin{align*} 1. & \quad \int_0^{\frac{\pi}{2}} \sin ^n x \, \mathrm{d} x=\int_0^{\frac{\pi}{2}} \cos ^n x \, \mathrm{d} x=\left\{\begin{array}{l} \frac{(n-1)}{n} \times \frac{(n-3)}{(n-2)} \times \cdots \times \frac{1}{2} \times \frac{\pi}{2}, \text { if } n=2,4,6, \cdots \\ \frac{(n-1)}{n} \times \frac{(n-3)}{(n-2)} \times \cdots \times \frac{2}{3}, \text { if } n=3,5,7, \cdots \end{array}\right. \\[0.2cm] 2. & \quad \int_0^{\frac{\pi}{2}}\sin^m x \cos^n x \, \mathrm{d}x \\[0.2cm] &= \frac{\left[ (m-1)(m-3)\cdots(2 \,\text{or}\, 1 )\right]\cdot \left[ (n-1)(n-3)\cdots(2 \, \text{or}\, 1) \right]}{(m+n)(m+n-2)(m+n-4)\cdots ( 2 \, \text{or}\, 1)}, \quad \text{If}\, m,n \in \mathbb{N} \text{ and both m and n are not even}\\[0.2cm] &= \frac{\left[ (m-1)(m-3)\cdots(2 \,\text{or}\, 1 )\right]\cdot \left[ (n-1)(n-3)\cdots(2 \, \text{or}\, 1) \right]}{(m+n)(m+n-2)(m+n-4)\cdots ( 2 \, \text{or}\, 1)}\cdot \frac{\pi}{2}, \quad \text{If}\, m,n \in \mathbb{N} \text{ and both m and n are even} \end{align*}

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Sanjivani wandre

15 April 2024 at 18:45

Very useful

Tukaram Thange

23 April 2024 at 11:05

Nice blog on list of integration formulae

Sujit Wandre

6 May 2024 at 15:46

I would highly recommend this blog to anyone looking to deepen their understanding of integration and explore its applications in diverse fields. It’s not just a resource; it’s a gateway to unlocking the profound intricacies of calculus.