Trigonometric Integration, Definite integral.

[Pinedas42]

Homework Statement

Use 2 or more substitutions to find the following integrals
hint : begin with u=cosx

Homework Equations

Integral 0--->pi/2 (cosxsinx)/swrt(cos²x+49 dx

The Attempt at a Solution

I'm still a little fuzzy on using multiple substitutions. From what I've read on the text and previous easier equations, it just means that there are multiple u=(something) that can work. Is that right?

so I tried u=cosx
du=-sinxdx

giving me


-1 * Integral 0--->pi/2 u/sqrt(u²+49) du


it's here that I am brickwalling. I really want to know how it works, so if you wouldn't mind a step by step process, I'd appreciate greatly.

 

[SammyS]

Homework Statement

Use 2 or more substitutions to find the following integrals
hint : begin with u=cosx

Homework Equations

Integral 0--->pi/2 (cosxsinx)/swrt(cos²x+49 dx

The Attempt at a Solution

I'm still a little fuzzy on using multiple substitutions. From what I've read on the text and previous easier equations, it just means that there are multiple u=(something) that can work. Is that right?

so I tried u=cosx
du=-sinxdx

giving me-1 * Integral 0--->pi/2 u/sqrt(u²+49) duit's here that I am brickwalling. I really want to know how it works, so if you wouldn't mind a step by step process, I'd appreciate greatly.

You should either change the limits of integration, or do the corresponding indefinite integration.

You have $\displaystyle \int \frac{u}{\sqrt{u^2+49}} du$.

Can you see a substitution which might work with your result?

(I can see two, either of which looks helpful.)

 

[Pinedas42]

So I beat at it until I solved it :D ( I don't give up dammit)

I put the integral into terms of u
so
u=cos(0)=1
u=cos(pi/2)=0so

integral 0-->1 u(u^2+49)^-1/2 du
I took the second sub of t=u^2+49
dt=2udu
to give
1/2 integral0-->1 (t)^-1/2 dt
1/2 * 2 (t)^1/2
giving the function
(u^2+49)^1/2 |0-->1
then using the fundamental theorem of calculus
[(1^2+49)^1/2]-[0^2+49]^1/2]
sqrt(50)-sqrt(49)
giving
-7+5sqrt(2)

:D

pretty stoked lol

 

[SammyS]

Excellent! (and welcome to PF !)

Another (very nice) subst. would have been to let t = √(u²+49) .

Try it, you might like it.