# Exponential and Logarithmic Functions

* Verify that the natural logarithm function defined as an integral has the same properties as the natural logarithm function earlier defined as the inverse of the natural exponential function.

Integrals of Exponential and Logarithmic Functions

Function| Integral|

lnx| x ∙ lnx - x + c|

logx| (x ∙ lnx - x) / ln(10) + c|

logax| x(logax - logae) + c|

ex| ex+c|

ek∙x| 1 / k ∙ ek∙x + c|

ax| ax / lna + c|

xn| 1 / (n+1) ∙ xn+1 + c, where |n|≠ 1 |

1/x = x-1| ln|x|+c|

√x = x1/2| 2/3 ∙ (√x)3 + c = 2/3 ∙ x3/2 + c, where c is a constant|

Example 1: Solve integral of exponential function ∫ex32x3dx Solution:

Step 1: the given function is ∫ex^33x2dx

Step 2: Let u = x3 and du = 3x2dx

Step 3: Now we have: ∫ex^33x2dx= ∫eudu

Step 4: According to the properties listed above: ∫exdx = ex+c, therefore ∫eudu = eu + c Step 5: Since u = x3 we now have ∫eudu = ∫ex3dx = ex^3 + c So the answer is ex^3 + c

Example 2: Integrate .

Solution: First, split the function into two parts, so that we get:

Trigonometric Identities and Ratio and Proportion

Trigonometric Identities

In mathematics, an "identity" is an equation which is always true. These can be "trivially" true, like "x =x" or usefully true, such as the Pythagorean Theorem's "a2 + b2 = c2" for right triangles. There are loads of trigonometric identities, but the following are the ones you're most likely to see and use. Ratio

The quantitative relation between two amounts showing the number of times one value contains or is contained within the othe Proportion

A part, share, or number considered in comparative relation to a whole.

Degrees| Radians|

360o| 2π|

180o| π|

90o| π/2|

60o| π/3|

45o| π/4|

30o| π/6|

1o| π/180|

180o/π| 1|

OBJECTIVES

* Set up and solve a proportion for a missing quantity.

* Solve applied problems using proportions.

* Find dimensions, areas, and volumes of similar...

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