How can you evaluate a definite integral using trigonometric substitution?

[doctordiddy]

Homework Statement

evaluate the definite integral ∫(0 to 3) dx/sqrt(25+x^2)

Homework Equations

The Attempt at a Solution

I first used substitution and set x=5tanθ, and dx=5tanθsecθdθ

then i wrote the integral as 5∫ tanθsecθdθ/sqrt(25(1+tan^2(θ))

after some simplification i got

∫tanθsecθdθ/secθ = ∫tanθ =tanθsecθ

I then used θ=arctan(x/5) from the original x substitution and my final solution looked like this

tan(arctan(3/5))sec(arctan(3/5))-tan(arctan(0))sec(arctan(0))

It is incorrect, can anyone let me know where i might have done something incorrectly? Thanks

 

[jedishrfu]

if x=5 tan(theta) then isn't dx=5sec^2(theta) dtheta

 

[doctordiddy]

if x=5 tan(theta) then isn't dx=5sec^2(theta) dtheta

oh you're right, but after solving the integral i still end up with tanθsecθ.. Unless i am doing something wrong in finding my θ values for x=3 and x=0?

 

[Dick]

oh you're right, but after solving the integral i still end up with tanθsecθ.. Unless i am doing something wrong in finding my θ values for x=3 and x=0?

No, you don't get tanθsecθ. And the integral of tanθ isn't even tanθsecθ. What are you doing?

 

[SammyS]

Homework Statement

evaluate the definite integral ∫(0 to 3) dx/sqrt(25+x^2)

Homework Equations

The Attempt at a Solution

I first used substitution and set x=5tanθ, and dx=5tanθsecθdθ

then i wrote the integral as 5∫ tanθsecθdθ/sqrt(25(1+tan^2(θ))

after some simplification i got

∫tanθsecθdθ/secθ = ∫tanθ =tanθsecθ

I then used θ=arctan(x/5) from the original x substitution and my final solution looked like this

tan(arctan(3/5))sec(arctan(3/5))-tan(arctan(0))sec(arctan(0))

It is incorrect, can anyone let me know where i might have done something incorrectly? Thanks

If you use your substitution to find $\displaystyle \ \int \frac{dx}{\sqrt{25+x*2}}\,,\$ you get $\displaystyle \ \int\sec(\theta)\,d\theta\ .$

The anti-derivative of sec(θ) is not tan(θ)sec(θ) . The derivative of sec(θ) is tan(θ)sec(θ) . You seem to have these confused.

A better substitution uses the hyperbolic sine, x = sinh(u) .

 

[doctordiddy]

If you use your substitution to find $\displaystyle \ \int \frac{dx}{\sqrt{25+x*2}}\,,\$ you get $\displaystyle \ \int\sec(\theta)\,d\theta\ .$

The anti-derivative of sec(θ) is not tan(θ)sec(θ) . The derivative of sec(θ) is tan(θ)sec(θ) . You seem to have these confused.

A better substitution uses the hyperbolic sine, x = sinh(u) .

oh right thanks, i used ∫secx=ln(secx+tanx) and it worked. Just a question about the hyperbolic sine, would you mind explaining it a bit to me or linking me to a good place that explains it? I don't think I've been taught it yet but one of the listed possible solutions has arccosh in it, and I am curious as to what exactly it is

 

[SammyS]

Here's a link to the Wikipedia page on hyperbolic functions.