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Selected answers to common questions:
Maximizing the volume of a box.
Maximizing the volume of a cylinder.
Volume of a tank.
What is a derivative?
- Integrals and Trig Functions [4/6/1996]
The integral from pi to zero of the square root of (1-sinX).
- Integrals in Polar Coordinates [04/15/2003]
How can I evaluate an expression like (sin(theta)*(constant)) for
theta = 0...2*Pi?
- Integrals versus Antiderivatives [02/24/2001]
What is the difference between an integral and an antiderivative?
- Integral tan(x)? [6/10/1996]
What is the antiderivative of the function tan(x)?
- Integral using Substitution [4/11/1996]
Evaluate: xdx / (x^2+1)^(1/2), given u^2 = x^2+1.
- Integrate Cos 2a [08/15/2001]
The answer given in the book for integrating cos 2a is sin 2a/2. Does
this follow a pattern?
- Integrating 1/(1+x^n) [11/14/2001]
Are there methods of integrating functions of the form 1/(1+x^n)? I have
solved 1/(1+x^4) and am wondering about solutions for higher powers of x.
- Integrating 1/(sinx)^6 dx [02/25/2002]
What is the integration of 1/(sinx)^6 dx?
- Integrating a Product of Cosines [05/04/2005]
Let n and m be natural numbers. My book says that the integral from
-pi to pi of the function f(x) = cos(nx) * cos(mx) is pi if n = m and
0 else. I can't figure out how to prove this.
- Integrating a Quadruple-Angle Trigonometric Expression or Two [01/13/2011]
A network engineer seeks help integrating a trigonometric expression with a cosine
term in its denominator (or is it the cotangent?). Using double-angle identities and a
substitution, Doctor Ali provides a boost.
- Integrating Exponentials [5/27/1995]
What is the integral of 5^x dx ?
- Integrating exp(-x^2) [11/09/2005]
It seems like all the ways of finding integral exp(-x^2) involve
making it a double integral and using polar coordinates. I was
wondering if there is any way to do it without making it a double
- Integrating with the Arc Length Formula [10/19/1998]
Can you help me integrate functions using the arclength formula, e.g.
x^(2/3)+y^(2/3)=1 and y=e^-x?
- Integrating X^x, Closed Form [10/31/1996]
How do you express the equation y = xcosx in terms of y?
- Integration [5/23/1996]
Integrate the following function: x^2 * e^(-2x^2)
- Integration [5/29/1996]
What does integral of (cos(x).x(cub).sinh(x)square).dx mean, and what's
- Integration [01/08/1998]
What is the integrated area if the anti-derivative is undefined?
- Integration [02/17/1999]
Integrate x*tan^2 x with respect to x.
- Integration [03/30/1999]
Integrate (2 - x^2)^4x dx.
- Integration by Partial Fractions [11/14/1996]
How do you integrate f(x) = x^2/[(x^4) + 1]?
- Integration by Parts [2/15/1995]
I need the integral from 0 to 1 of X^4*exp(-x/2). Can you help?
- Integration by Parts [12/8/1995]
How do you Integrate: x^n/e^x ?
- Integration by Parts [5/24/1996]
How do you integrate x^x?
- Integration by Parts [01/07/1997]
How do you solve Int [sin (ln x)] dx?
- Integration by Parts [05/27/1997]
What is the integral of sec^3(x)?
- Integration by Parts [04/08/1999]
How does one find the integral of [(sec x)^3 dx]?
- Integration by Parts [05/02/1999]
Find the general integral of x*ln(x)-1.
- Integration by Parts [04/10/2008]
How can I integrate e^x(x+1)/(x+2)^2 ?
- Integration Hints [02/23/1999]
Can you help me integrate (cos[x])^4?
- Integration of Natural Logs [02/12/1998]
Why does the natural log of x equal the integral of 1/t dt from 1 to x?
Why is INT[(1/t)dt] from 1 to x the natural log of x, or why was it
defined this way?
- Integration of Sin(x^2) [11/10/1997]
I have been given the solution in the form of Frensel's Sin, but it
explains nothing about how it was integrated. I am not looking for an
equation, I am looking for a reason!
- Integration of Trigonometric Function by Substitution [03/18/1998]
Integrate [tan 2x dx] by u-substitution.
- Integration of Trigonometric Functions [05/28/1999]
How can I integrate sin(x)sin(2x).dx ?
- Integration Over a Complicated Region [11/1/1996]
Given a region defined by five relations simultaneously, find its area.
- Integration Problem [2/22/1995]
How do you integrate the function: (tan 2x)^3 dx, in terms of x?
- Integration Questions [03/09/2001]
Calculate the area of the finite region bounded by the curves y=x^2 and
- Integration Trick [8/13/1996]
If f(x)=1/sin(x), what is the integral of f(x)?
- An Integration with Substitution [06/06/1998]
I'm having trouble with the following integration. Is substitution needed
- Intriguing Limit [04/30/1997]
Why is the following true: lim x --> oo (1-1/x)^x = 1/e?
- Introduction to Line Integrals [02/08/2004]
C is the broken path line that goes from (1,1,2) to (2,0,-1) by
following a line parallel to the x axis from (1,1,2) to (2,1,2) then
parallel to the y axis from (2,1,2) to (2,0,2) then parallel to the z
axis from (2,0,2) to (2,0,-1). Find int_C [ f(x,y,z)*dx + g(x,y,z)*dy
+ h(x,y,z)*dz ].