# Integrals: Why Use Arctan Formula?

In summary, the conversation discusses solving an integral using the arctan formula and u-substitution. The individual providing the solution initially forgot to include the 2 when differentiating, but later corrected it in an edit.
i don't see why the following integral:$$\int\frac{dx}{\sqrt{x}(x + 1)}$$ uses the arctan formula. i know how to solve integrals. i just don't see why or how you can rewrite it using the arctan formula. Isn't the arctan formula used for integrals of the form: $$\int\frac{du}{a^2 + u^2}=\frac{1}{a}arctan\frac{u}{a}$$

$$\int \frac{dx}{\sqrt{x}(x+1)}$$
you have to use u-substitution.
so, you have:
$$u = \sqrt{x}~~~du=\frac{1}{2 \sqrt{x}$$
now, here's where it gets tricky:
$$x=u^2$$
now, you can substitute back in.
$$\int \frac{du}{u^2+1}$$
now, just integrate and plug u back in, and you are done.

*edit* it isn't let me post du...well, i can't see it from my comp, but du = 1/2sqrt(x)dx

Last edited:
thanks i see now.

He missed the 2 when differentiating $x=u^{2}$.

Daniel.

$$u = \sqrt{x}$$
$$du = \frac{1}{2}x^{-\frac{1}{2}}dx$$

? What is wrong with that?

Nothing,just that u didn't take it into account in the post i was referring to.

Daniel.

hehe, yea i left out the 2...so its half the integral when you work it out. this computer at work was frustrating me, so i forgot it. (my excuse )

## 1. What is the Arctan formula used for in integrals?

The Arctan formula is used to solve integrals involving the inverse tangent function. It helps to find the antiderivative of a function that contains the inverse tangent function.

## 2. How is the Arctan formula derived?

The Arctan formula is derived from the fundamental theorem of calculus, which states that the derivative of the antiderivative of a function is equal to the original function. The formula is also derived using trigonometric identities and integration by substitution.

## 3. What are the advantages of using the Arctan formula in integrals?

The Arctan formula simplifies the integration process and allows for the evaluation of integrals that may not be solvable using other methods. It also helps to find the exact solution to integrals involving inverse tangent functions.

## 4. Are there any limitations to using the Arctan formula?

The Arctan formula can only be used for integrals involving the inverse tangent function. It cannot be used for other types of integrals and may not always give the most efficient solution.

## 5. Are there any real-life applications of the Arctan formula?

Yes, the Arctan formula has many real-life applications in fields such as physics, engineering, and economics. It is used to solve problems involving angles, curves, and rates of change. It is also used in the calculation of areas and volumes of curved objects.

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