Need help solving this indefinite integral via integration by substitution

[lo2]

Homework Statement

Calculate the following integral:

\int{\frac{\sqrt{x+1}}{x+5}dx} \ , x ≥ 1

By using the following substitution:

t=\sqrt{x+1}

Homework Equations

Well using the integration by substitution formula.

The Attempt at a Solution

So I have t=\sqrt{x+1} \Rightarrow \frac{dx}{dt}=\frac{1}{2\sqrt{x+1}} \Rightarrow dx=dt \frac{1}{2\sqrt{x+1}}

But I do not really have this "piece" inside the integral, so I guess you have to apply some special technique which I have forgotten about. So can someone help me?

 

[Curious3141]

Homework Statement

Calculate the following integral:

\int{\frac{\sqrt{x+1}}{x+5}dx} \ , x ≥ 1

By using the following substitution:

t=\sqrt{x+1}

Homework Equations

Well using the integration by substitution formula.

The Attempt at a Solution

So I have t=\sqrt{x+1} \Rightarrow \frac{dx}{dt}=\frac{1}{2\sqrt{x+1}} \Rightarrow dx=dt \frac{1}{2\sqrt{x+1}}

But I do not really have this "piece" inside the integral, so I guess you have to apply some special technique which I have forgotten about. So can someone help me?

Homework Statement

Homework Equations

The Attempt at a Solution

You need to put everything purely in terms of t, if possible. You can rearrange t = \sqrt{x+1} to give you x + 5 in terms of t

The expression for dx can also be expressed purely in terms of t. Rewrite it this way, then substitute it back into the integral and you should find a fairly simple integral wrt t after things cancel out.

 

[HallsofIvy]

Homework Statement

Calculate the following integral:

\int{\frac{\sqrt{x+1}}{x+5}dx} \ , x ≥ 1

By using the following substitution:

t=\sqrt{x+1}

Homework Equations

Well using the integration by substitution formula.

The Attempt at a Solution

So I have t=\sqrt{x+1} \Rightarrow \frac{dx}{dt}=\frac{1}{2\sqrt{x+1}} \Rightarrow dx=dt \frac{1}{2\sqrt{x+1}}

Well, that last is dt/(2t) isn't it? Also, since t= \sqrt{x+ 1}, t^2= x+ 1 so x+ 5= t^2+ 4

Your integral is
\int \frac{t}{t^2+ 4}\left(\frac{dt}{2t}\right)

But I do not really have this "piece" inside the integral, so I guess you have to apply some special technique which I have forgotten about. So can someone help me?

 

[lo2]

Well, that last is dt/(2t) isn't it? Also, since t= \sqrt{x+ 1}, t^2= x+ 1 so x+ 5= t^2+ 4

Your integral is
\int \frac{t}{t^2+ 4}\left(\frac{dt}{2t}\right)

Hey thanks a lot for the help! Well those substitutions do make sense, but my problem is that the resulting integral is as far as I can see not that easy :/

So would I have to use integration by substitution once again to solve this, or integration by parts?

I must admit that my integration technique is a wee bit rusty...

 

[HallsofIvy]

What? The final integral is
\int \frac{dt}{t^2+ 4}
which is a basic integral.

(What is the derivative of arctan(x)?)

 

[supernova1387]

If you use t=sqrt[x+1], you will get :

I=2∫t²/(t²+4)dt . Now change the nominator, separate and integrate:


I=2∫t²/(t²+4)dt=2∫(t²+4-4)/(t²+4)dt=

2∫[1-4/(t²+4)]dt=2[∫1dt-4∫dt/(t²+4)]= 2 Sqrt[1 + x] - 4 ArcTan[Sqrt[1 + x]/2]

 

[supernova1387]

What? The final integral is
\int \frac{dt}{t^2+ 4}
which is a basic integral.

(What is the derivative of arctan(x)?)

So you mean derivative of arctan can be written as Sqrt[x + 1]/(x + 5)?

 

[Mark44]

No, HallsOfIvy is not saying that d/dx(arctan(x)) = sqrt(x+1)/(x + 5). He was giving a hint to the OP as to how to proceed.

 

[lo2]

Ok I have come up with the following solution: (please correct me if I am wrong!)

We have this integral:

\int{\frac{\sqrt{x+1}}{x+5}dx}

Where

t=\sqrt{x+1} \Leftrightarrow t^2+4=x+5

And where

\frac{dx}{dt}=\frac{1}{2t} \Leftrightarrow dx=\frac{1}{2t}dt

This gives us the following integral:

\int{\frac{t}{t^2+4}\cdot \frac{1}{2t}dt} = \int{\frac{1}{2t^2+8}dt} = \int{\frac{1}{\frac{1}{4}t^2+1}dt}

The anti derivative of this is:

Arctan{(\frac{1}{4}t)} +c = Arctan{(\frac{1}{4}\sqrt{x+1})} +c

 

[HallsofIvy]

Almost. The anti-derivative of 1/(u^2+ 1) is arctan(u)+ C but with
1/(x^2/4+ 1), u is NOT "x/4".

 

[lo2]

Yeah ok well, I was afraid it might be so.

But I am not sure I really can see how you are going to use the Arctan connection... So could you or someone else perhaps be very kind and explain me how to use this fact and how to rearrange the equation so that you can use this connection.

Would be most appreciated! ;)

 

[supernova1387]

Ok I have come up with the following solution: (please correct me if I am wrong!)

We have this integral:

\int{\frac{\sqrt{x+1}}{x+5}dx}

Where

t=\sqrt{x+1} \Leftrightarrow t^2+4=x+5

And where

\frac{dx}{dt}=\frac{1}{2t} \Leftrightarrow dx=\frac{1}{2t}dt

This gives us the following integral:

\int{\frac{t}{t^2+4}\cdot \frac{1}{2t}dt} = \int{\frac{1}{2t^2+8}dt} = \int{\frac{1}{\frac{1}{4}t^2+1}dt}

The anti derivative of this is:

Arctan{(\frac{1}{4}t)} +c = Arctan{(\frac{1}{4}\sqrt{x+1})} +c

Did you actually look at what I wrote for you? I wrote the final answer for you too.

You have t²+4=x+5 so you get: dx/dt=2t → dx=2tdt
so you have t² in your nominator and not t.

 

[lo2]

If you use t=sqrt[x+1], you will get :

I=2∫t²/(t²+4)dt . Now change the nominator, separate and integrate:


I=2∫t²/(t²+4)dt=2∫(t²+4-4)/(t²+4)dt=

2∫[1-4/(t²+4)]dt=2[∫1dt-4∫dt/(t²+4)]= 2 Sqrt[1 + x] - 4 ArcTan[Sqrt[1 + x]/2]

Well no I do not really understand this to be honest...

How do you get:

2 \int{\frac{t^2}{t^2+4} dt}

From:

\int{\frac{t}{t^2+4}\cdot \frac{1}{2t}dt}

As far as I can see those are not the same :/

 

[eumyang]

Let's go back to your first attempt:

By using the following substitution:

t=\sqrt{x+1}

...

So I have t=\sqrt{x+1} \Rightarrow \frac{dx}{dt}=\frac{1}{2\sqrt{x+1}} \Rightarrow dx=dt \frac{1}{2\sqrt{x+1}}

The expressions after the first right arrow aren't correct. If
t=\sqrt{x+1},
then
dt=\frac{dx}{2\sqrt{x+1}},
So
dx=2\sqrt{x+1} dt and
dx=2t dt

When you substitute into
\int{\frac{\sqrt{x+1}}{x+5}dx}
you get
\int{\frac{t}{t^2 + 4}\cdot 2t dt}, or
\int \frac{2t^2}{t^2 + 4}

 

[Curious3141]

I have to apologise for my earlier post, which may have been misleading. Nothing cancels out, and you're left with:

2\int \frac{t^2}{t^2 + 4}dt

as Eumyang wrote.

To resolve that, rewrite the integrand as \frac{t^2 + 4 - 4}{t^2 + 4} = 1 - \frac{4}{t^2 + 4} = 1 - \frac{1}{{(\frac{t}{2})}^2 + 1}.

So the integral becomes:

2\int 1 - \frac{1}{{(\frac{t}{2})}^2 + 1} dt

That can be integrated either by observation, or if you want, by a simple substitution of u = \frac{t}{2} to clarify the form.

 

[lo2]

Ok now I have the solution solution: (please correct me if I am wrong!)

We have this integral:

\int{\frac{\sqrt{x+1}}{x+5}dx}

Where

t=\sqrt{x+1} \Leftrightarrow t^2+4=x+5

And where

\frac{dt}{dx}=\frac{1}{2t} \Leftrightarrow dx=2tdt

This gives us the following integral:

\int{\frac{t}{t^2+4}\cdot 2tdt} = 2\int{\frac{t^2}{t^2+4}dt} = 2\int{\frac{t^2+4-4}{t^2+4}dt}=2\int{1-\frac{1}{(\frac{t}{2})^2+1}dt}

The anti derivative of this is, where I use the substitution u=\frac12:

\frac{du}{dt}=\frac{1}{2} \Leftrightarrow du=\frac{1}{2}dt


2\int{1-\frac{1}{(\frac{t}{2})^2+1}dt}=2\int{2(1-\frac{1}{(\frac{t}{2})^2+1}\frac12)dt}=4\int{1-\frac{1}{(u)^2+1}du}=4(u-Arctan(u))=2\sqrt{x+1}-4Arctan(\frac12 \sqrt{x+1})+c

So I guess this is how it should be solved?

And is Arctan=cot^{-1} ?

 

[Curious3141]

Ok now I have the solution solution: (please correct me if I am wrong!)

We have this integral:

\int{\frac{\sqrt{x+1}}{x+5}dx}

Where

t=\sqrt{x+1} \Leftrightarrow t^2+4=x+5

And where

\frac{dt}{dx}=\frac{1}{2t} \Leftrightarrow dx=2tdt

This gives us the following integral:

\int{\frac{t}{t^2+4}\cdot 2tdt} = 2\int{\frac{t^2}{t^2+4}dt} = 2\int{\frac{t^2+4-4}{t^2+4}dt}=2\int{1-\frac{1}{(\frac{t}{2})^2+1}dt}

The anti derivative of this is, where I use the substitution u=\frac12:

\frac{du}{dt}=\frac{1}{2} \Leftrightarrow du=\frac{1}{2}dt


2\int{1-\frac{1}{(\frac{t}{2})^2+1}dt}=2\int{2(1-\frac{1}{(\frac{t}{2})^2+1}\frac12)dt}=4\int{1-\frac{1}{(u)^2+1}du}=4(u-Arctan(u))=2\sqrt{x+1}-4Arctan(\frac12 \sqrt{x+1})+c

So I guess this is how it should be solved?

A couple of typos in your post, but your solution is correct, so I believe your method is, too.

And is Arctan=cot^{-1} ?

No arctan is the inverse function of tan (tangent). It's often depicted as \tan^{-1}, and the exponent does not refer to the reciprocal, it refers to the inverse function. The cotangent (cot) is the reciprocal of tan. The inverse function of cot is called arccotangent, depicted as arccot, or \cot^{-1}, following the same convention.

I know you're asking the question because of what Wolfram outputted. The \cot^{-1} term in the output is referring to the arccotangent function. This is a trigonometric identity (all the -1 exponents refer to inverse functions):

\tan^{-1}{y} = \cot^{-1}{\frac{1}{y}}

so you should easily be able to see that your answer is the same as Wolfram's.

 

[lo2]

How come you are so clever?

 

[Curious3141]

How come you are so clever?

If you're talking to me, well, I'm not, really, but it's nice of you to say so.

 

[lo2]

If you're talking to me, well, I'm not, really, but it's nice of you to say so.

I was indeed referring to you! But well thanks a lot for the help :)

Btw when solving integrals which includes division of some kind, e.g. two functions divided by each other. Is there a general approach to that, or does it depend on the concrete example?

 

[Curious3141]

I was indeed referring to you! But well thanks a lot for the help :)

Btw when solving integrals which includes division of some kind, e.g. two functions divided by each other. Is there a general approach to that, or does it depend on the concrete example?

It depends on the example. There's no one size fits all solution.

 

[lo2]

It depends on the example. There's no one size fits all solution.

Ok what about these kind of integrals:

\int{\frac{1}{ax+b}dx}

 

[Curious3141]

Ok what about these kind of integrals:

\int{\frac{1}{ax+b}dx}

This form is very simple to integrate.

\int {(ax+b)}^{-1} dx = \frac{1}{a}\ln |ax + b| + c

You should be able to "immediately" integrate something this simple by observation alone. But if you're having trouble seeing why the answer is the way it is, try the substitution y = ax + b, or, equivalently, x = \frac{y - b}{a}.

 

[lo2]

Yeah well that is fairly simple...

But also I made a typo actually I was referring to integrals of this kind, where as far I know you cannot just multiplicate the derivative on, as it is not merely constant.

The kind of integrals I was referring to, look like this:

\int{\frac{1}{ax^2+b}dx}

 

[SammyS]

Yeah well that is fairly simple...

But also I made a typo actually I was referring to integrals of this kind, where as far I know you cannot just multiplicate the derivative on, as it is not merely constant.

The kind of integrals I was referring to, look like this:

\int{\frac{1}{ax^2+b}dx}

Factor a out of the denominator giving:

\displaystyle \int{\frac{1}{ax^2+b}dx}

\displaystyle =\frac{1}{a}\int{\frac{1}{x^2+b/a}dx}

\displaystyle =\frac{1}{a}\int{\frac{1}{x^2+\left(\sqrt{b/a}\right)^2}dx}
​

Letting \displaystyle c=\sqrt{\frac{b}{a}} gives you:

\displaystyle =\frac{1}{a}\int{\frac{1}{x^2+c^2}dx}