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use the substitution u=2+ln(x) to show that int(e,1(ln(x)/x(2+ln(x)^2))dx)=p+ln(q) , where p and q are rational numbers.

So u=2+lnx, therefore du/dx=1/x , we can work out the new upper and new lower limit by substitute in e and 1 into 2+lnx , and we get 2+ln(e)=3 , 2+ln(1)=2Rearrange the differential we get dx=xdu , substitute...
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Solve the quadratic inequality: x^2 - 5x + 4 < 0

x 2 -5x+4 &lt;0First we ignore the inequality and try to solve the equation x 2 -5x+4=0, which we do via factorising (x-4)(x-1)=0. x = 4 or x=1We draw the graph using our solution, going through the points o...
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Answered by Hariz H. • Maths tutor
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Find two positive numbers whose sum is 100 and whose product is a maximum.

Call the two numbers x and y. The constraint is that x + y =100, and we need to maximise A=xy. Rearrange the constraint to y = 100 - x, and substitute into the product equation. A = x(100-x) = 100x - x 2 Dif...
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Solve the integral: int(x^3+4x^2+sinx)dx.

First, since there are terms with different factors of x summing together, we can separate these into three individual integrals as follows:int(x 3 dx) + 4int(x 2 dx) + int(sinx dx)Then using the rules of in...
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A curve C has equation: y = x^2 − 2x − 24x^1/2, x > 0; Find (i) dy/dx (ii) d^2y/dx^2

i) let dy/dx = ans, where ans = y, where you multiply the coefficients of each constant of x by the power and reduce the power by 1 in the equation ytherefore dy/dx = 2x - 2 - 12x -1/2 ii) repeat the process...
CH
Answered by Charles H. • Maths tutor
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