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).
    

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

Indefinite Integrals

If F(x) is a function whose derivative F'(x) = f(x) on certain interval of the x-axis, then F(x) is called the anti-derivative of indefinite integral f(x). When we integrate the differential of a function we get that function plus an arbitrary constant. In symbols we write
 

Problem 648
For the cantilever beam loaded as shown in Fig. P-648, determine the deflection at a distance x from the support.