
MathMake a substitution `y = 2x,` then `dy = 2 dx` and the limits of integration for `y` are from `0` to `3.` The integral becomes the table one: `int_0^3 1/(1+y^2) (dy)/2 = 1/2 (tan)^(1)(y) _(y=0)^3...

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MathMake a substitution `x = 3y,` then `dx = 3 dy` and the limits of integration for y are from 0 to 1/18. The integral becomes a table one: `int_0^(1/18) (9 dy)/(sqrt(99y^2)) = int_0^(1/18) (3...

MathRecall that the indefinite integral is denoted as: `int f(x) dx =F(x)+C` There properties and basic formulas of integration we can apply to simply certain function. For the problem `int...

MathThe derivative of a function with respect to t is denoted as f'(t). The given function:` f(x) = arcsin(t^2) ` is in a form of a inverse trigonometric function. Using table of derivatives, we have...

MathBy definition, if the function F(x) is the antiderivative of f(x) then we follow the indefinite integral as `int f(x) dx = F(x)+C` where: f(x) as the integrand F(x) as the...

MathIndefinite integral are written in the form of` int f(x) dx = F(x) +C` where: f(x) as the integrand F(x) as the antiderivative function C as the arbitrary constant known...

Math`y= log_3(x^23x)` The derivative formula of a logarithm is: `d/(dx) [log_a (u)] = 1/(ln(a) * u) * (du)/(dx)` Applying this formula, the derivative of the function will be: `(dy)/(dx) = d/(dx)...

Math`y=log_4(5x + 1)` The derivative formula of a logarithm is: `d/(dx) [log_a (u)] = 1/(ln(a) * u) * (du)/(dx)` Applying that formula, the derivative of the function will be: `(dy)/(dx) = d/(dx)[...

Math`f(t) = 3^(2t)/t` To take the derivative of this function, use the quotient rule `(u/v)'= (v*u'  u*v')/v^2`. Applying that, f'(t) will be: `f'(t) = (t * (3^(2t))'  3^(2t)*(t)')/t^2` `f'(t) =...

MathThe derivative of a function f at a point x is denoted as y' = f'(x). There are basic properties and formula we can apply to simplify a function such as the Product Rule provides the formula:...

MathRecall that the derivative of a function f at a point x is denoted as `y' = f'(x)` . There basic properties and formula we can apply to simplify a function. For the problem `y = x(6^(2x)),` we...

Math`y=6^(3x4)` The derivative formula of an exponential function is: `d/(dx) (a^u) = ln(a) * a^u * (du)/dx` Applying this formula, the derivative of a function will be: `(dy)/(dx) = d/(dx)...

Math`y=5^(4x)` The derivative formula of an exponential function is: `d/(dx) (a^u) = ln(a) * a^u * (du)/(dx)` Applying this formula, the derivative of the function is: `(dy)/(dx) = d/(dx)(5^(4x))`...

MathThe derivative of f(x) with respect to x is denoted a f'(x). The given function f(x) = 3^(4x) is in exponential form which means we can apply the basic integration formula: `d/(dx)(a^u) =...

MathBy definition, the derivative of f(x) with respect to x is denoted a f'(x) where `f'(x) = lim (f(x+h) f(x))/h ` as` h>0` . Instead of using the limit of difference quotient, we may apply the...

MathTo simplify the logarithmic equation: `log_5(sqrt(x4))=3.2` , recall the logarithm property: `a^((log_(a)(x))) = x` . When a logarithm function is raised by the same base, the log cancels out...

MathTo solve a logarithmic equation, we may simplify or rewrite it using the properties of logarithm. For the given problem `log_(3)(x^2)=4.5` , we may apply the property: `a^((log_(a)(x))) = x` The...

MathIn solving a logarithmic equation, we may simplify using logarithm properties. Recall the logarithm property: `a^((log_(a)(x))) = x` . When we raise the log with the same base, the "log" will...

Math`log_2 (x1) =5` To solve, convert the equation to exponential form. Take note that if a logarithmic equation is in the form `y = log_b (x)` its equivalent exponential equation is `x= b^y` So...

MathProblem:`3(5^(x1))=86` ` ` To simplify, we divide both sides by 3: `(3(5^(x1)))/3=(86)/3` `5^(x1)=(86)/3` ` ` ` ` Take the "log" on both sides to apply the logarithm property:...

MathFor exponential equation:`2^(3z)=625` , we may apply the logarithm property: `log(x^y) = y * log (x)` . This helps to bring down the exponent value. Taking "log" on both sides:...

MathFor exponential equation: `5^(6x)= 8320` , we may apply the logarithm property: `log(x^y) = y * log (x).` This helps to bring down the exponent value. Taking "log" on both sides:...

MathProblem:` 3^(2x)=75` is an exponential equation. To simplify, we need to apply logarithm property: `log(x^y) = y*log(x)` to bring down the exponent that is in terms of x. Taking "log" on both...

Math`log_3(x) + log_3(x  2) = 1` The logarithms at the left side have the same base. So express the left side with one logarithm only using the rule `log_b (M) + log_b (N) = log_b(M*N` ). `log_3(x *...

Math`x^2x=log_5 (25)` First, simplify the right side of the equation. To do so, factor 25. `x^2  x = log_5 (5^2)` Then, apply the logarithm rule `log_b (a^m) = m * log_b (a)` . `x^2  x = 2 * log_5...

Math`log_b (27) = 3` To solve, convert this to exponential equation. Take note that if a logarithmic equation is in the form `y = log_b (x)` its equivalent exponential equation is: `x = b^y` So...

Math`log_3 (x) = 1` To solve, convert this to exponential equation. Take note that if a logarithmic equation is in the form `y = log_b (x)` its equivalent exponential equation is: `x = b^y` So...

Math`log_3(1/81) = x` To solve this, consider the base and argument of the logarithm. The base of the logarithm is 3. And the argument has 81 which is multiple of 3. Factoring 81, the equation...

Math`log_10 (1000) = x` To solve, express the 1000 in terms of 10. `log_10 (10 * 10 * 10)=x` `log_10 (10^3) =x` To simplify the left side, apply the logarithm rule `log_b (a^m) = m*log_b(a)` . `3*...

MathTake note that if a logarithmic equation is in the form `y = log_b (x)` its equivalent exponential equation is: `x = b^y` Thus, converting `log_3 (1/9) = 2` to exponential equation, it becomes...

MathTake note that if a logarithmic equation is in the form `y=log_b(x)` its equivalent exponential equation is `x=b^y` Thus, converting `log_10 (0.01) =2` to exponential equation, it becomes:...

MathTake note that if the exponential equation is in the form `y=b^x` its equivalent logarithmic form is `x=log_b (y)` Thus, converting `27^(2/3)=9` to logarithmic form, it becomes `2/3=log_27 (9)`

MathTake note that if an exponential equation is in the form `y = b^x` its equivalent logarithmic form is `x=log_b(y)` Thus, converting `2^3=8` to logarithmic form, it becomes: `log_2(8)=3`

Math`log_a (1/a)` To simplify this, apply the negative exponent rule for `1/a` . `=log_a (a^(1))` Then, apply the logarithmic rule `log_b(x^m)=m*log_b(x)` . `= 1 * log_a (a)` Take note that when the...

Math`log_7 (1)` Take note that if the argument of a logarithm is 1, its resulting value is zero `(log_b(1) = 0)` . `= 0` Therefore, `log_7 (1) = 0` .

Math`log_27 (9)` To evaluate, let this expression be equal to y. `y = log_27 (9)` Then, convert this equation to exponential form. Take note that if a logarithmic equation is in the form `y = log_b...

Math`log_2 (1/8)` To evaluate this, consider the base and argument of the logarithm. The base of the logarithm is 2. And the 8 can be expressed in terms of factor 2. So factoring 8, the expression...

MathHello! It is obvious that there is no linear formula exactly connecting these x's and y's, if we consider the slopes between neighbor points: `(155  134)/(31  30) = 21,` `(165  155)/(33  31) =...

MathHello! Suppose that the two numbers are both >=2 (if one of the numbers is 1, the problem is trivial). Then both have its unique decomposition into prime factors. If we find this decomposition,...

MathParent functions are the most basic, or "simplest" form of a given function with no transformations placed upon them. For example, `y=4x^2+2x+5` is a quadratic function. The parent is: `y=x^2` The...

MathAn operation follows the Commutative property if changing the order of the numbers in the operation does not change the outcome. Thus, addition (and multiplication) are commutative (i.e., a+b=b+a...

MathFor a stem and leaf plot, the number to the left of the bar is generally the stem and the numbers to the right of the bar are the leaves. The stem is the first digit(s) and the leaves are the next...

MathTo do this, the longest rod would be the diagonal of the box. Now, this is different than the diagonal of one side of the box. I am talking about the diagonal that would go through the middle of...

MathWell, at some point, all subjects can be related. For instance, you could use math quite easily in social sciences, studying peoples and populations, for instance. The "variable" would be how you...

MathThis is a typical problem involving percents. There are a few different way to approach such problems. Recall that percent is a numerator (top) of a fraction with the denominator (bottom) equal to...

MathThis all depends, on all kinds of factors. For instance, even if each plant is planted separately, away from each other, they may be so far away from each other than one may be more/less sun...

MathIn order to factor this expression, first open all parenthesis by distributing the term in front of the parenthesis. The given expression will then become `a^2b + ab^2 + b^2c +bc^2 + ac^2 + a^2c +...

MathWhat does the variable n in mathematics usually mean? The variable can mean many things. 1) Maybe the variable is used in an equation only to represent an unknown, such as: `10 = 6n+4` 2) The...

Math`y=3ln(6t+1)` To take the derivative of this, refer to formula: `d/(dx) (ln u) = 1/u * (du)/dx ` Applying that, the derivative of the function will be: `d/(dt)(y) = d/(dt)[3ln(6t+1)]`...
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