Recent content by whkoh

Notice that \frac{d}{dx}\left(\frac{x^2}{2}\right)=x So it is actually \int{f'\left(x\right)e^{f\left(x\right)}=e^{f\left(x\right)}+c

Qn. By using a suitable substituition, find \int{\frac{1}{x\sqrt{1+x^n}}dx} I haven't encountered this specific type of question before, so I went to use the obvious substitution u^2=1+x^n, getting: 2u=n x^{n-1} \frac{dx}{du}\Leftrightarrow \frac{dx}{du}=\frac{2u}{n} x^{1-n} Hence...

By writing (1+x) as \frac{1}{2}\left[1+\left( 1+2x\right) \right] or otherwise, show that the coefficient of x^n in the expansion of \frac{\left(1+x\right)^n}{\left(1+2x\right)^2} in ascending powers of x is \left(-1\right)^n\left(2n+1\right). -- I've tried expressing (1+x)^n as...

Thanks for your help!

The numbers x and y satisfy 0 < x \leq a^2, 0 < y \leq a^2, xy \geq a^2 where a \geq 1. By sketching suitable graphs or otherwise, show that x + y \geq 2a and x \leq a^{2}y \leq a^{4}x --- I don't know what to sketch (tried x \leq 1, y \leq 1, xy \leq 1), so I tried algebraic methods...

By using the Sine formula in triangle ABC, show that: \frac{a+b}{c} = \frac{cos\frac{A-B}{2}}{sin\frac{c}{2}}. I've tried: \frac{2 sin C}{c} = \frac{sin A}{a} + \frac{sin B}{b} \frac{2 sin C}{c} = \frac{b sin A + a sin B}{a+b} \frac{a+b}{c} = \frac{b sin A + a sin B}{2 sin C} \frac{a+b}{c} =...

Great, thanks for the help!

Solve for x in terms of a, the inequality: \mid x^2 - 3ax + 2a^2 \mid < \mid x^2 + 3ax - a^2 \mid where x \in \mathbb{R}, a \in \mathbb{R}, a \neq 0 Squaring both sides, I get x^4 - 6ax^3 + 13a^2 x^2 - 12a^3 x + 4a^4 < x^4 + 6ax^3 + 7a^2 x^2 - 6a^3 x + a^4 12ax^3 - 6a^2 x^2 + 6a^3 x -...

Many thanks for your help. I assumed that the lamp had a height and that's why I got everything wrong. Now I understand... Thanks!

The book is using the ratio 100*1000/3600 [=27 7/9] and not the approximation 27.7 m/s.

A lamp is located on the ground 10 m from a building. A man 1.8 m tall walks from the light toward the building at a rate of 1.5 m s⁻¹. What is the rate at which the man's shadow on the wall is shortening when he is 3.2 m from the building ? Give your answer correct to two decimal places...

Thanks for your help :smile:

Check the range of x: 0 < x < 360 180 < 2x + 180 < 900

Well, manipulating RHS gives \tan \left (\frac{\pi}{4} + \frac{x}{2} \right ) = \frac{\tan\frac{\pi}{4}+\tan\frac{x}{2}}{1-\tan\frac{\pi}{4}\tan\frac{x}{2}} =\frac{1+\tan\frac{x}{2}}{1-\tan\frac{x}{2}} and applying half angle to LHS gives \frac{1+\sin x}{\cos x}...

Prove that \sec x + \tan x = \tan \left (\frac{\pi}{4} + \frac{x}{2}\right ) I've got to \sec x + \tan x = \frac{1+\sin x}{\cos x} and then I was stuck. Tried half angle but it didn't seem to work. Help please.