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High School Archive

Browse High School Calculus
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Selected answers to common questions:
    Chain rule.
    Maximizing the volume of a box.
    Maximizing the volume of a cylinder.
    Volume of a tank.
    What is a derivative?

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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 integral?

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 the answer?

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 y=sqrt(x)...

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 here?

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 ].

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