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Solve the integral int(sin^2(x))dx

Knowing that 1-2sin 2 (x) = cos(2x), substitute sin 2 (x) as: sin 2 (x) = (1-cos(2x))/2Use the linearity of the integral to split it into: Int(1/2) - Int(cos(2x)/2)dxWhich is in the end:0.5x - 0.25 sin(2x) +...
MM
Answered by Maks M. • Maths tutor
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Consider f (x) = logk (6x - 3x 2 ), for 0 < x < 2, where k > 0. The equation f (x) = 2 has exactly one solution. What is the value of k?

There are two essential tricks to grasp in this question. Firstly, since the equation has only one solution, the Discriminant that will be required would equal 0. Secondly, since we are given f(x) = 2 we can...
JS
Answered by Jaroslav S. • Maths tutor
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Solve the equation log2(x + 3) + log2(x - 3) = 4

log2(x + 3) + log2(x - 3) = 4log2 [(x + 3)(x - 3)] = 4x^2 - 9 = 16x^2 = 25x = 5
Answered by • Maths tutor
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Prove by induction that 7^(8n+3) + 2 is divisible by 5, where n is a natural number.

We need to first consider the base case, n=0. (We consider the smallest possible value of n as the base case.)7^(8 0+3) + 2 = 343 + 2 = 345 = 69 * 5, so the statement is true for n=0. Assume true for n = k, ...
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Answered by Mingke P. • Maths tutor
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Solve for x in the following equation: e^x + 10e^(-x) = 7

First of all, we bring the 7 to the left side of the equation to get: e x -7+10e -x =0. Then, by multiplying both sides of the equation by e x , we can get an equation in the form of a quadratic equation: e ...
LB
Answered by Leonardo B. • Maths tutor
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