In this blog post, I will provide a comprehensive list of integration formulas to help you master this important topic.
Basic Integrals
\(
\begin{align*}
1. &\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) \\
2. &\int \frac{1}{x} \, dx = \ln|x| + C \\
3. &\int e^x \, dx = e^x + C \\
4. &\int a^x \, dx = \frac{a^x}{\ln a} + C \quad (a > 0) \\
5. &\int \sin x \, dx = -\cos x + C \\
6. &\int \cos x \, dx = \sin x + C \\
7. &\int \sec^2 x \, dx = \tan x + C \\
8. &\int \csc^2 x \, dx = -\cot x + C \\
9. &\int \sec x \tan x \, dx = \sec x + C \\
10. &\int \csc x \cot x \, dx = -\csc x + C \\
11. &\int \frac{1}{\sqrt{1-x^2}} \, dx = \sin^{-1}x + C \\
12. &\int \frac{1}{1+x^2} \, dx = \tan^{-1}x + C \\
13. &\int \frac{1}{|x|\sqrt{x^2-1}} \, dx = \sec^{-1}|x| + C
\end{align*}
\)
Integrals of Hyperbolic Functions
\(
\begin{align*}
1. &\int \sinh x \, dx = \cosh x + C \\
2. &\int \cosh x \, dx = \sinh x + C \\
3. &\int \text{sech}^2 x \, dx = \tanh x + C \\
4. &\int \text{csch}^2 x \, dx = -\coth x + C
\end{align*}
\)
Inverse Trigonometric Integrals
\(
\begin{align*}
1. &\int \sin^{-1}x \, dx = x\sin^{-1}x + \sqrt{1-x^2} + C \\
2. &\int \cos^{-1}x \, dx = x\cos^{-1}x – \sqrt{1-x^2} + C \\
3. &\int \tan^{-1}x \, dx = x\tan^{-1}x – \frac{1}{2}\ln(1+x^2) + C
\end{align*}
\)
Special Integrals
\(
\begin{align*}
1. &\int \ln x \, dx = x\ln x – x + C \\
2. &\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C \\
3. &\int \frac{1}{\sqrt{a^2 – x^2}} \, dx = \sin^{-1}\left(\frac{x}{a}\right) + C \\
4. &\int \frac{1}{x^2 – a^2} \, dx = \frac{1}{2a} \ln\left|\frac{x-a}{x+a}\right| + C \\
5. &\int \sqrt{a^2 – x^2} \, dx = \frac{x}{2}\sqrt{a^2 – x^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right) + C
\end{align*}
\)
Trigonometric Integrals
\(
\begin{align*}
1. &\int \sin^2 x \, dx = \frac{x}{2} – \frac{\sin 2x}{4} + C \\
2. &\int \cos^2 x \, dx = \frac{x}{2} + \frac{\sin 2x}{4} + C \\
3. &\int \tan^2 x \, dx = \tan x – x + C \\
4. &\int \sin mx \sin nx \, dx = \frac{\sin(m-n)x}{2(m-n)} – \frac{\sin(m+n)x}{2(m+n)} + C \quad (m \neq n) \\
5. &\int \sin mx \cos nx \, dx = -\frac{\cos(m-n)x}{2(m-n)} – \frac{\cos(m+n)x}{2(m+n)} + C
\end{align*}
\)
Reduction Formulas
\(
\begin{align*}
1. &\int \sin^n x \, dx = -\frac{\sin^{n-1} x \cos x}{n} + \frac{n-1}{n} \int \sin^{n-2} x \, dx \\
2. &\int \cos^n x \, dx = \frac{\cos^{n-1} x \sin x}{n} + \frac{n-1}{n} \int \cos^{n-2} x \, dx
\end{align*}
\)
