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

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

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

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,
 

The General Power Formula as shown in Chapter 1 is in the form
 

The General Power Formula
Logarithmic Functions
Exponential Functions
Trigonometric Functions
Trigonometric Transformation
Inverse Trigonometric Functions

Evaluate the following:

Example 4: $\displaystyle \int \sqrt{x^3 + 2} \,\, x^2 \, dx$

Example 5: $\displaystyle \int \dfrac{(3x^2 + 1) \, dx}{\root 3\of {(2x^3 + 2x + 1)^2}}$

Example 6: $\displaystyle \int (1 - 2x^2)^3 \, dx$
 

Evaluate the following integrals:

Example 1: $\displaystyle \int \dfrac{2x^3+5x^2-4}{x^2}dx$

Example 2: $\displaystyle \int (x^4 - 5x^2 - 6x)^4 (4x^3 - 10x - 6) \, dx$

Example 3: $\displaystyle \int (1 + y)y^{1/2} \, dy$
 

The definite integral of f(x) is the difference between two values of the integral of f(x) for two distinct values of the variable x. If the integral of f(x) dx = F(x) + C, the definite integral is denoted by the symbol
 

The quantity F(b) - F(a) is called the definite integral of f(x) between the limits a and b or simply the definite integral from a to b. It is called the definite integral because the result involves neither x nor the constant C and therefore has a definite value. The numbers a and b are called the limits of integration, a being the lower limit and b the upper limit.
 

Integration Formulas

In these formulas, u and v denote differentiable functions of some independent variable (say x) and a, n, and C are constants.
 

1. The integral of the differential of a function u is u plus an arbitrary constant C (the definition of an integral).
    

   $$\displaystyle \int du = u + C$$