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Calculus
The branch of mathematics that deals with the study of continuously changing quantities, with the use of limits and the differentiation and integration of functions of one or more variables, is called Calculus. Calculus analyzes aspects of change in processes or systems that can be modeled by functions. The English physicist, Isaac Newton, and the German mathematician, G. W. Leibniz, working independently, developed calculus during the 17th century.
25,068 Questions
What is the integral of one over the square root of forty-nine minus x squared?
Very sorry, I showed you how to take the derivative of the stated problem. I leave that below, however to take the integral you really just need your table of integrals.
The reference integral from my table (in my old calculus book) is:
integral of 1 / [ a2 - x^2]1/2 = sin-1|x / a| + c
a2 = 49 => a=7
x2 = x^2 => x = x
And your answer is:
sin
-1(x / 7) + c
If you are not allowed to use the table of integrals, the other alternative is trigonometric substitution.
Assume our integral is:
The integral of dx / sqrt ( a2 - x2)
Let x = a sin (θ)
dx = a cos (θ) d
θ
<-- this is just the derivative of x
θ = sin-1 (x/a) <-- solve the first equation for theta
So the integral transforms into:
the integral of ( a cos (θ)
d
θ
) / sqrt ( a2 -
a2sin2(θ) )
factor out an a2 in the root:
( a cos (θ)
d
θ
) / sqrt (
a
2 (1
-
sin2(θ) ) )
use the trig identity: sin2
(θ) + cos2
(θ) = 1
which can be rewritten:
1
-
sin2(θ) =
cos2
(θ)
and therefore:
( a cos (θ)
d
θ
) / sqrt (
a
2(
cos2
(θ)
) )
which simplifies to:
( a cos (θ)
d
θ
) /
( a cos (θ) )
which leaves us with the integral of
d
d
θ
it's now easy to solve and we get:
θ + C
plugin theta (from above):
sin
-1(x/a) + C
Plus in your a and x from the original equation, and you are good to go.
-----
Bonus: Here is how you find the derivative of this, since I already solved this on accident:
First, rewrite the expression to kill the fraction:1/(49-x2)1/2 =(49-x2)-1/2
Now, we can use u-substitution.
Let u = 49-x2
u` = -2x
Take thederivativeof u-1/2
(u-1/2)`= [ -u-3/2 ]/ 2
Now multiply by u`.
( [ -u-3/2]/ 2 ) ( -2x ) = (the 2's and negatives cancel)
x [ u-3/2] =
x / u3/2
Now plug u back in (remember: u = 49-x2)
x / u3/2 = x / (49-x2)3/2
The answer is:
x / (49-x2)3/2
Can you solve the equation 4x squared plus 7 equals 5?
Yes, but the solution is imaginary if you have learned that
4x^2 + 7 = 5
4x^2 = -2
x^2 = -1/2
x = +/-0.707i where i = square root of -1
log(x6) = log(x) + log(6) = 0.7782*log(x)
log(x6) = 6*log(x)
What does the graph of x minus x squared look like?
the x-axis... obviously! the x-axis... obviously!
What is x in this problem 2x 39.x?
Unfortunately, the browser used by Answers.com for posting questions is rubbish and incapable of accepting mathematical symbols. This means that we cannot see the mathematically critical parts of the question. We are, therefore unable to determine what exactly the question is about and so cannot give a proper answer to your question.
Why can you not divide a number by 0?
You can't divide something like 5 by nothing. It's irrational, because the number zero represents nothing, and that it is just a symbol recognizing that nothing is there. It's almost like an invisible thing, because 0 represents invisibility, because it is the only symbol justifying nothing in the numerical system.
What is eighty-two percent written as a fraction in simplest form?
Expressed as a proper fraction in its simplest form, 82 percent is equal to 41/50 or forty-one fiftieths.
Asymptote of x² times e to the power of x?
As x tends to negative infinity, the expression is asymptotically 0.
What capacity are 1997 Chevy Cheyenne gas tanks?
The capacity of a 1997 Chevy Cheyenne gas tank is 128 liters. Or one could say it is 32 gallons.
Weight is measured in Newtons, not kg. A kilogram measure mass.
The density is approx 2.7 grams/cubic cm.
How do you round 87 and frame the first digit?
you should round it to the nearest tenth and box the first number
∛(x4) √(x5)
= x4/3x5/2
= x8/6x15/6
= x23/6
How do you find the volume of the solid between x-y equals 1 and y equals x2-4x plus 3?
Both of those equations represent two-dimensional graphs, so they're not the boundaries
of a solid.
They do, however, enclose an area between them, in the interval between x=1 and x=4.
Evaluating the area takes a little bit of calculus. It's "beyond the scope" of a usual WikiAnswer,
I don't think you want to see that explained in detail, and I don't feel like it right now anyway.
What is 20x plus 80y equals 0 in y equals mx plus b?
20x + 80y = 0
Subtract 20x from each side of the equation:
80y = -20x
Divide each side by 80:
y = -1/4 x
m = -1/4
b = 0
