Exponential and Logarithmic Functions
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
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