Saturday

April 18, 2015

April 18, 2015

Posted by **MS** on Thursday, October 31, 2013 at 8:15am.

- Calculus -
**Count Iblis**, Thursday, October 31, 2013 at 11:27amsin(x) = sin[2(x/2)] =

2 sin(x/2) cos(x/2)

Draw a right triangle with one angle equal to x/2. If you make the length of the side opposite to that angle equal to

t = tan(x/2) then the length of the side side orthogonal to it that connects to that angle will be equal to 1, because ratio of the two sides must be equal to tan(x/2) = t.

This means that the length of the hypotenuse is equal to:

h = sqrt(1+t^2)

and you have that:

sin(x/2) = t/sqrt(1+t^2)

cos(x/2) = 1/sqrt(1+t^2)

Therefore:

sin(x) = 2 t/(1+t^2)

And:

x = 2 arctan(t) ------>

dx = 2dt/(1+t^2)

This gives:

dx/(4-5 sin x)

2 dt/(1+t^2) 1/[4 - 10 t/(1+t^2)] =

dt/[2+2 t^2 -5t] =

dt/[2(t-1/2)(t-2)] =

dt/(t-1/2) * 1/[2* (1/2 - 2)] +

dt/[(t-2)] * 1/[2* (2-1/2)]

Integrating gives:

-1/3 Log|t-1/2| + 1/3 Log|t-2| =

1/3 Log|[tan(x/2) - 2]/[tan(x/2) -1/2]|

**Answer this Question**

**Related Questions**

calculus - 1) Integrate Cos^n(x) dx 2) Integrate e^(ax)Sinbx dx 3) Integrate (...

Calculus - s=dx/(4+5cos x). By using t-substitution, i.e. t=tan(x/2) we get ...

Calculus - Integrate (1/2)sin(x^(1/2))dx. I've tried using u-substitution, with ...

Calculus - Calc length of arc of y=ln(x) from x=1 to x=2 ---- So far: Definite ...

calculus - Integrate x/(x^2 + 4) dx via trig substitution and by u=x^2+4 ...

Calculus - Find the integral, using the method of substitution where necessary. ...

calculus - integrate x*(x+1)^9dx I believe the substitution method is the proper...

Integral calculus - Please can anyone help with the following problems - thanks...

Maths Calculus Derivatives - Find the first derivative for the following ...

Trig - Given: cos u = 3/5; 0 < u < pi/2 cos v = 5/13; 3pi/2 < v < ...