# Solving this integral with u substitution

• karush
In summary: So this means that the OP should read:Homework Statement: Evaluate ##\displaystyle\int_{0}^{3}\frac{x+3}{\sqrt{x^{2}+1}}dx+5##Relevant Equations: u SubstitutionIn summary, Wolffram Alpha showed that the substitution u = x^{3/2} does give a closed form solution to the equation, but that another substitution might be necessary in order to solve for u.
karush
Gold Member
MHB
Homework Statement
u subst
Relevant Equations
u subst
Evaluate ##\displaystyle\int_{0}^{3}\frac{x+3}{\sqrt{x^{3}+1}}dx+5##
W|A returned 11.7101
ok subst is probably just one way to solve this so
##u=x^{3}+1 \quad du= 3x^2##

Last edited:
What did you try? what is w|A?

Wolffram Alpha
u subst

Last edited:
Please show your work

Orodruin said:
Please show your work
I did what I could in the OP

You did nothing in the OP apart from just stating a substitution. What does that give you? Where do you get stuck? Please be specific.

ok I can't see how this subst would play out
or do I need to go somewhere elae for help
$$u=x^3+1\quad du=3x^2 \quad (u-1)^{1/3}=x$$
this doesn't render

Last edited by a moderator:
karush said:
this doesn't render
It does render properly now that I changed your single-$delimiters to double-$ delimiters.

karush said:
or do I need to go somewhere elae for help
If you expect us to do your work for you without you showing any effort, then yes. If you are willing to put in some effort, then you will get great help here at PF.

I don't think an ordinary substitution by itself will do the trick. Something to try is 1) the substitution ##u = x^{3/2}##, followed by 2) a trig substitution. The first substitution turns the denominator to ##\sqrt{u^2 + 1}##, which suggests a trig substitution. I worked it through part way, but didn't complete my work, so I'm not sure that this will bear fruit.

berkeman said:
It does render properly now that I changed your single-$delimiters to double-$ delimiters. If you expect us to do your work for you without you showing any effort, then yes. If you are willing to put in some effort, then you will get great help here at PF.

sorry there is a typo in the OP it should be ##x^2## not ##x^3##

i have already solved the problem by expansion

karush said:
sorry there is a typo in the OP it should be ##x^2## not ##x^3##

i have already solved the problem by expansion
So this means that the OP should read:
karush said:
Homework Statement: Evaluate ##\displaystyle\int_{0}^{3}\frac{x+3}{\sqrt{x^{2}+1}}dx+5##

Relevant Equations: u subst

ok subst is probably just one way to solve this so
The indefinite integral, ##\displaystyle \int \frac{x+3}{\sqrt{x^{2}+1}}dx##, does have a closed form solution,

Break that into the sum of integrals: ##\displaystyle \int \frac{x}{\sqrt{x^{2}+1}}dx + 3\int \frac{1}{\sqrt{x^{2}+1}}dx## .

The first can be handled by a relatively simple substitution.

The second can be done with:
1) a trig substitution, the result of which may require knowing ##\int \sec \theta \ d\theta ##.

2) a hyperbolic function substitution .

3) knowledge of the derivative of the inverse hyperbolic sine function.

PhDeezNutz, Mark44, vanhees71 and 1 other person
Mahalo every one

## What is u substitution?

U substitution is a technique used in calculus to simplify integrals by substituting a new variable, u, for the original variable in the integrand. This allows for easier integration and can help solve more complex integrals.

## When should I use u substitution?

U substitution is most commonly used when the integrand contains a function and its derivative, as well as when the integrand contains a polynomial expression and a radical expression.

## How do I choose the u value?

The u value should be chosen based on the derivative in the integrand. It should be a function that, when differentiated, will result in the remaining terms in the integrand. In some cases, trial and error may be necessary to find the best u value.

## What is the process for solving an integral with u substitution?

The process for solving an integral with u substitution involves choosing a u value, substituting it into the integrand, simplifying the integral, and then integrating the new expression. After integration, the final answer should be in terms of u, and then the u value should be replaced with the original variable.

## Are there any common mistakes to avoid when using u substitution?

One common mistake when using u substitution is forgetting to replace the u value with the original variable in the final answer. It is also important to carefully differentiate the u value and substitute it correctly into the integrand. Additionally, it is important to check that the limits of integration are also adjusted when using u substitution.

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