# TBI's question at Yahoo Answers (Indefinite integral)

• MHB
• Fernando Revilla
In summary, the integral of (e^sqrt(x) + 1) with respect to x is equal to 2e^sqrt(x)(sqrt(x)-1)+x+C. This can be derived using the integration by parts method and substitution. The final result is a boxed equation that can be used to solve for the integral.
Fernando Revilla
Gold Member
MHB
Hello TBI,

Denote $I=\int(e^{\sqrt{x}}+1)\;dx$, then $I=\int e^{\sqrt{x}}\;dx+x+C$. If $t=\sqrt{x}$, then $dt=\dfrac{dx}{2\sqrt{x}}=\dfrac{dx}{2t}$ so: $$\int e^{\sqrt{x}}\;dx=2\int te^{t}\;dt$$ Using the integration by parts method:

\begin{aligned}\left \{ \begin{matrix}u=t\\dv=e^tdt\end{matrix}\right.& \Rightarrow \left \{ \begin{matrix}du=dt\\v=e^t\end{matrix}\right.\\& \Rightarrow \int te^{t}\;dt=te^t-\int e^t\;dt\\&=\sqrt{x}e^{\sqrt{x}}-e^{\sqrt{x}}\end{aligned} As a consequence: $$\boxed{\;I=\displaystyle\int(e^{\sqrt{x}}+1)\;dx=2e^{\sqrt{x}}(\sqrt{x}-1)+x+C\;}$$

## 1. What is a TBI?

A TBI, or traumatic brain injury, is a type of brain injury caused by a sudden physical force to the head or body, such as from a fall, sports injury, or car accident. It can result in a wide range of symptoms, including changes in cognition, behavior, and physical abilities.

## 2. What is an indefinite integral?

An indefinite integral is a mathematical concept used in calculus to find the antiderivative of a function. It represents the inverse operation of differentiation and is typically written as an integral sign (∫) followed by the function to be integrated and the differential variable. It is used to solve problems involving rates of change and accumulation.

## 3. How are TBI's treated?

The treatment for a TBI depends on the severity and type of injury. It may include rest, medication, physical therapy, occupational therapy, speech therapy, and counseling. In more severe cases, surgery may be necessary. Each treatment plan is tailored to the individual's specific needs and goals for recovery.

## 4. What is the difference between a definite and indefinite integral?

The main difference between a definite and indefinite integral is the presence or absence of limits of integration. A definite integral has specific limits, while an indefinite integral does not. This means that a definite integral will give a numerical value, while an indefinite integral will result in a function.

## 5. Can a person fully recover from a TBI?

The extent of recovery from a TBI varies greatly and depends on many factors, such as the severity of the injury and the individual's health and age. Some people may fully recover, while others may experience long-term effects. It is important to seek appropriate treatment and support to maximize recovery and improve quality of life.

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