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  • Math
    Complete the square at the denominator: `x^4 + 2x^2 + 2 = (x^2)^2 + 2x^2 + 1 + 1 = (x^2 + 1)^2 + 1.` Now we see the substitution `y = x^2 + 1,` then `dy = 2x dx,` and the integral becomes `int...

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  • Math
    By completing the square and making simple substitution, we will reduce this integral to a table one. `-x^2-4x = -(x^2 + 4x + 4) + 4 = -(x+2)^2 + 4 = 4 - (x+2)^2.` Now make a substitution `y =...

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  • Math
    Make 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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  • Math
    Make 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(9-9y^2)) = int_0^(1/18) (3...

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  • Math
    Recall 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...

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  • Math
    The given function `f(x) = arctan(sqrt(x))` is in a inverse trigonometric form. The basic derivative formula for inverse tangent is: `d/(dx) arctan(u) = ((du)/(dx))/sqrt(1-u^2)` . Using...

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  • Math
    The 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...

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  • Math
    `arccos(1)` Let this expression be equal to y. `y = arccos(1)` Rewriting it in terms of cosine function, the equation becomes: `cos(y) = 1` Base on the Unit Circle Chart, cosine is 1 at angles 0...

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  • Math
    `arccos(1/2)` Let this expression be equal to y. `y = arccos(1/2)` Rewriting this in terms of cosine function the equation becomes: `cos(y) =1/2` Base on the Unit Circle Chart, cosine is 1/2 at...

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  • Math
    `arcsin(0)` Let this expression be equal to y. `y =arcsin(0)` Re-writing this equation in terms of sine function, it becomes: `sin (y) = 0` Base on the Unit Circle Chart (see attached figure), sine...

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  • Math
    `arcsin(1/2)` Let this expression be equal to y. `y =arcsin(1/2)` Rewriting it in terms of sine function, the equation becomes: `sin(y) = 1/2` Base on the Unit Circle Chart (see attached figure),...

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  • Math
    To evaluate the integral: `int_(-4)^(4) 3^(x/4) dx` , we follow the formula based from the First Fundamental Theorem of Calculus: `int_a^bf(x)dx=F(b)- F(a)` wherein f is a continuous and F is...

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  • Math
    Recall the First Fundamental Theorem of Calculus: If f is continuous on closed interval [a,b], we follow: `int_a^bf(x)dx` = F(b) - F(a) where F is the anti-derivative of f on [a,b]. This...

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  • Math
    By 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...

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  • Math
    Indefinite integral are written in the form of` int f(x) dx = F(x) +C` where: f(x) as the integrand F(x) as the anti-derivative function C as the arbitrary constant known...

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  • Math
    Derivative of a function h with respect to t is denoted as h'(t). The given function: `h(t) = log_5(4-t)^2` is in a form of a logarithmic function. From the derivative for logarithmic functions,...

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  • Math
    `y= log_3(x^2-3x)` 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)...

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  • 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)[...

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  • 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) =...

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  • Math
    The 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:...

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  • Math
    Recall 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...

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  • Math
    `y=6^(3x-4)` 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)...

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  • 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))`...

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  • Math
    The 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) =...

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  • Math
    By 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...

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  • Math
    To simplify the logarithmic equation: `log_5(sqrt(x-4))=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...

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  • Math
    To 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...

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  • Math
    In 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...

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  • Math
    `log_2 (x-1) =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...

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  • Math
    Problem:`3(5^(x-1))=86` ` ` To simplify, we divide both sides by 3: `(3(5^(x-1)))/3=(86)/3` `5^(x-1)=(86)/3` ` ` ` ` Take the "log" on both sides to apply the logarithm property:...

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  • Math
    For exponential equation:`2^(3-z)=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:...

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  • Math
    For 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:...

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  • Math
    Problem:` 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...

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  • 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 *...

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  • Math
    `x^2-x=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...

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  • 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...

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  • 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...

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  • 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...

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  • 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*...

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  • Math
    Take 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...

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  • Math
    Take 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:...

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  • Math
    Take 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)`

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  • Math
    Take 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`

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  • 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...

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  • 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` .

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  • 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...

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  • 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...

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  • Math
    Hello! 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) =...

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  • Math
    Hello! 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,...

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