How to Solve an Integral with Trigonometric Functions and Exponents?

[rich_machine]

Homework Statement

evaluate the integral from 0 to e^x of sin(tan^-1(e^(sqrt (x)))dx

Homework Equations

all trig function such as sin=opp/hypo, cos=adj/hypo, tan oppo/adj

The Attempt at a Solution

I believe the integral can be written as the integral from 0 to e^x of (e^sqrt(x))/(sqrt(1+e^(2sqrt(x))).

Anyone know how to solve this problem?

 

[Mark44]

Homework Statement

evaluate the integral from 0 to e^x of sin(tan^-1(e^(sqrt (x)))dx

Homework Equations

all trig function such as sin=opp/hypo, cos=adj/hypo, tan oppo/adj

The Attempt at a Solution

I believe the integral can be written as the integral from 0 to e^x of (e^sqrt(x))/(sqrt(1+e^(2sqrt(x))).

Anyone know how to solve this problem?

Start by sketching a right triangle and label the sides and an acute angle θ so that tan(θ) = e^√x. Find sin(θ).

 

[rich_machine]

using that i can get integral from 0 to e^x of (e^sqrt(x))/(sqrt(1+e^(2sqrt(x)))

how should I continue

 

[rich_machine]

Start by sketching a right triangle and label the sides and an acute angle θ so that tan(θ) = e^√x. Find sin(θ).

using that I can find integral from 0 to e^x of (e^sqrt(x))/(sqrt(1+e^(2sqrt(x)))

how should I continue

 

[Mark44]

Where is dx? If you are in the habit of ignoring it and omitting it, you will come to grief, and things will become very difficult.

Here's the integral that you are reporting in post #4:
$$ \int_0^{e^x} \frac{e^{\sqrt{x}}~dx}{\sqrt{1 + e^{2\sqrt{x}}}}$$

It's not a very good practice to have the same variable in one or both limits of integration as in the "dummy" variable of the integrand. Let's change that.
$$ \int_{t = 0}^{t = e^x} \frac{e^{\sqrt{t}}~dt}{\sqrt{1 + e^{2\sqrt{t}}}}$$

That looks better.

To make forward probress, an ordinary substitution might be called for here.

 

[Dick]

Where is dx? If you are in the habit of ignoring it and omitting it, you will come to grief, and things will become very difficult.

Here's the integral that you are reporting in post #4:
$$ \int_0^{e^x} \frac{e^{\sqrt{x}}~dx}{\sqrt{1 + e^{2\sqrt{x}}}}$$

It's not a very good practice to have the same variable in one or both limits of integration as in the "dummy" variable of the integrand. Let's change that.
$$ \int_{t = 0}^{t = e^x} \frac{e^{\sqrt{t}}~dt}{\sqrt{1 + e^{2\sqrt{t}}}}$$

That looks better.

To make forward probress, an ordinary substitution might be called for here.

I don't think any simple substituition will deal with that, it's a very nasty integral. I think you need nonelementary functions to deal with it, like polylogarithms. I suspect rich_machine isn't telling us the whole question. Do you want to find the derivative of that integral? Finding the derivative of an integral doesn't require finding the integral first.

 

[Mark44]

I don't think any simple substituition will deal with that, it's a very nasty integral.

Mostly, I was looking for directions to go, and it was bothersome that with the substitution I was heading towards, there wasn't anything to make up du.

II think you need nonelementary functions to deal with it, like polylogarithms. I suspect rich_machine isn't telling us the whole question. Do you want to find the derivative of that integral?

This thought did cross my mind.

Finding the derivative of an integral doesn't require finding the integral first.

 

[rich_machine]

Thank you dick and mark. I believe my calc professor has assigned a problem well beyond the scope of the class. Most likely he created the problem himself at a moments notice and did not realize the difficulty of the integral.

 

[haruspex]

I assume the tan^-1 means arctan, not cotan.

My inclination is to get rid of the arctan first, setting it to theta:

Spoiler

$\int\sin(\theta).dx$ where $x = ln(2\tan(\theta))$, so $dx = \frac 1{\sin(\theta)\cos(\theta)}dθ$.

Looks easy from there.