Math Help - Calculus with Integration

1. Calculus with Integration

Hi I got urgent problems with this assignment involving integration

One is substituting t into the equation.

Limits [1, 10] Integrate (1*x^-1/2)* log*x^1/2 dx
Using x = t^2.

Firstly I worked out dx = 2t and substitute x and dx to get
(1/t)*logt 2t

Then I think that integrating that will get me

1/2[Ln|t|*1/t] with limits 1 and 10 but I got huge doubts at this stage.

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And this one involves Trigionomic Substitution.

And Limits ["pi"/4, "pi"/2] Integrate dx/sinx where you use t = tan (x/2)
Only thing cae to mind was identities like tan(x/2) = sin(x/2)/cos(x/2) but didnt get far on this one either

Could you guys help me out pls?

Thx

2. Originally Posted by NeloAngelo
Hi I got urgent problems with this assignment involving integration

One is substituting t into the equation.

Limits [1, 10] Integrate (1*x^-1/2)* log*x^1/2 dx
Using x = t^2.

Firstly I worked out dx = 2t and substitute x and dx to get
(1/t)*logt 2t

Then I think that integrating that will get me

1/2[Ln|t|*1/t] with limits 1 and 10 but I got huge doubts at this stage.

==============================================

And this one involves Trigionomic Substitution.

And Limits ["pi"/4, "pi"/2] Integrate dx/sinx where you use t = tan (x/2)
Only thing cae to mind was identities like tan(x/2) = sin(x/2)/cos(x/2) but didnt get far on this one either

Could you guys help me out pls?

Thx
after sub you should get

$\int_{1}^{\sqrt{10}}\frac{1}{t}\log(t)(2t)dt=2\int _{1}^{\sqrt{10}}\log(t)dt$

try integration by parts and note: $\log{t}=\frac{ln(t)}{ln(10)}$