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

Exponential Functions | Fundamental Integration Formulas

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

Logarithmic Functions | Fundamental Integration Formulas

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 General Power Formula | Fundamental Integration Formulas

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

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

 

Chapter 2 - Fundamental Integration Formulas

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

4 - 6 Examples | Indefinite Integrals

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

1 - 3 Examples | Indefinite Integrals

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

Definite Integral

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
 

$\displaystyle \int_a^b f(x) \, dx = F(b) - F(a)$

 

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.
 

Properties of Integrals

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

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
 

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