The General Power Formula

The General Power Formula as shown in fundamental theorems is in the form
 

$\displaystyle \int u^n \, du = \dfrac{u^{n+1}}{n+1} + C; \,\,\, n \neq -1$

 

Thus far integration has been confined to polynomial functions. Although the power formula was studied, our attention was necessarily limited to algebraic integrals, so that further work with power formula is needed. The power formula can be used to evaluate certain integrals involving powers of the trigonometric functions.
 

Logarithmic Functions

The limitation of the Power Formula $\displaystyle \int u^n \, du = \dfrac{u^{n + 1}}{n + 1} + C$, is when $n = -1$; this makes the right side of the equation indeterminate. This is where the logarithmic function comes in, note that $\displaystyle \int u^{-1} \, du = \displaystyle \int \frac{du}{u}$, and we can recall that $d(\ln u) = \dfrac{du}{u}$. Thus,
 

$\displaystyle \int \dfrac{du}{u} = \ln u + C$

 

The formula above involves a numerator which is the derivative of the denominator. The denominator $u$ represents any function involving any independent variable. The formula is meaningless when $u$ is negative, since the logarithms of negative numbers have not been defined. If we write $u = -v$ so that $du = -dv$, then we have
 

$\displaystyle \int \dfrac{du}{u} = \int \dfrac{-dv}{-v} = \int \dfrac{dv}{v} = \ln v + C = \ln (-u) + C$

 

When negative numbers are involved, the formula should be considered in the form
 

$\displaystyle \int \dfrac{du}{u} = \ln | u | + C$

 

The integral of any quotient whose numerator is the differential of the denominator is the logarithm of the denominator.
 

Exponential Functions

There are two basic formulas for the integration of exponential functions.

1. $\displaystyle \int a^u \, du = \dfrac{a^u}{\ln a} + C, \,\, a > 0, \,\, a \neq 1$

2. $\displaystyle \int e^u \, du = e^u + C$
 

Where
$u$ = function, say $f(x)$
$a$ = constant (example: 3, π, sin 30°, √7)

$e = {\underset{x \to \infty}\lim} \left( x + \dfrac{1}{x} \right)^x = {\underset{x \to \infty}\lim} (1 + x)^{1/x}$

$e = 2.71828\,1828\,4590\,4...$
 

Trigonometric Functions

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
 

Inverse Trigonometric Functions

In applying the formula (Example: Formula 1 below), it is important to note that the numerator du is the differential of the variable quantity u which appears squared inside the square root symbol. We mentally put the quantity under the radical into the form of the square of the constant minus the square of the variable.
 
1. $\displaystyle \int \dfrac{du}{\sqrt{a^2 - u^2}} = \arcsin \, \dfrac{u}{a} + C, \,\,\, a > 0$

2. $\displaystyle \int \dfrac{du}{a^2 + u^2} = \dfrac{1}{a}\arctan \, \dfrac{u}{a} + C$

3. $\displaystyle \int \dfrac{du}{u\sqrt{u^2 - a^2}} = \dfrac{1}{a} {\rm arcsec} \, \dfrac{u}{a} + C$
 

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