Trigonometric Functions | Fundamental Integration Formulas

Basic Formulas

1. $\displaystyle \int \sin u \, du = -\cos u + C$

2. $\displaystyle \int \cos u \, du = \sin u + C$

3. $\displaystyle \int \sec^2 u \, du = \tan u + C$

4. $\displaystyle \int \csc^2 u \, du = -\cot u + C$

5. $\displaystyle \int \sec u \, \tan u \, du = \sec u + C$

6. $\displaystyle \int \csc u \, \cot u \, du = -\csc u + C$
 

Formulas Derived from Logarithmic Function

7. $\displaystyle \int \tan u \, du = \ln (\sec u) + C = -\ln (\cos u) + C$

8. $\displaystyle \int \cot u \, du = \ln (\sin u) + C$

9. $\displaystyle \int \sec u \, du = \ln (\sec u + \tan u) + C$

10. $\displaystyle \int \csc u \, du = \ln (\csc u - \cot u) + C = -\ln (\csc u + \cot u) + C$
 
The six basic formulas for integration involving trigonometric functions are stated in terms of appropriate pairs of functions. An integral involving $\sin x$ and $\tan x$, which the simple integration formula cannot be applied, we must put the integrand entirely in terms of $\sin x$ and $\cos x$ or in terms of $\tan x$ and $\sec x$. Notice that these formulas are reverse formulas in Differential Calculus.
 
The formulas derived from trigonometric function can be traced as follows:
 
$\displaystyle \int \tan u \, du$

$\,\,\,\,\,\,\,\,\, = \displaystyle \int \dfrac{\sin u \, du}{\cos u}$

$\,\,\,\,\,\,\,\,\, = -\displaystyle \int \dfrac{-\sin u \, du}{\cos u}$

$\,\,\,\,\,\,\,\,\, = -\ln (\cos u) + C$            → Formula

$\,\,\,\,\,\,\,\,\, = \ln (\cos u)^{-1} + C$

$\,\,\,\,\,\,\,\,\, = \ln \left(\dfrac{1}{\cos u} \right) + C$

$\,\,\,\,\,\,\,\,\, = \ln (\sec u) + C$            → Formula
 

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