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Integrate cos(log(x)) dx

There are several approaches to integrating this function, thus making this a valuable exercise for students needing to develop fluency with the various techniques of integration and practise dealing with in...
SB
5071 Views

Integrate the function f(x) = x ln (x) over the interval [1,e].

This problem can be solved using integration by parts (IBP), although using IBP multiple times: BASIC IBP METHOD: given an expression integral(u dv) where u and v are functions of a common variable x and 'dv...
JH
5294 Views

Compute the derivative of arcsin(x).

To compute the derivative of arcsin(x) we use the fact that it is the inverse of sine. Write y=arcsin(x). We want dy/dx. Taking sin on both sides yields sin(y)=x. Use implicit differentiation to differentiat...
JP
7267 Views

The circle c has equation x^2+ y ^2=1 . The line l has gradient 3 and intercepts the y axis at the point (0, 1). c and l intersect at two points. Find the co-ordinates of these points.

The equation for a line is y(x)=ax+b.Because it intersects the y axis at x=0 at y=1y(x)=b=1a is equal to the gradient of the function so the line is given as y(x)=3x+1In order to get the two intersections we...
CB
3660 Views

By use of matrices uniquely solve the following system of equations, justifying each step of the calculation: 3x-7y=6, 5y-2x=-3.

The first thing observe is that we are being asked to use a specific approach in order to solve for both x and y , hence we should not approach this question via elimination of a single variable, though it i...
KM
3970 Views