# Solving Integrals: \int\frac{xdx}{1+x^4}

• Echo 6 Sierra
In summary, an integral is a mathematical concept used to find the area under a curve on a graph. To solve an integral, one must use algebraic manipulation and integration techniques to find an antiderivative and evaluate it over a given range. The general process for solving integrals involves identifying the type, finding the antiderivative, applying techniques, and evaluating. The purpose of solving integrals is to find the total value of a function over a given range, which has many real-world applications. Common integration techniques include substitution, integration by parts, partial fractions, and trigonometric substitution.
Echo 6 Sierra
OK, for the bad mamma-jamma:
$$\int\frac{xdx}{1+x^4}$$ I picked $$u=x^2$$ and made du=2xdx which makes 1/2du=xdx.

Now I have: $$\displaystyle{\frac{1}{2}}$$$$\int\frac{du}{1+u^2}$$

If the almighty green-bottled lager from Holland has been good to me, I get:

$$\frac{1}{2} \arctan x^2+c$$

Is this: a) correct, b)good enough or c)can this be tweaked?

So help me, if I get at least a B in Calculus I will be near Spiritual Creaminess.

Yes it's correct.

First of all, great job on solving the integral! Your approach of using u-substitution and picking u=x^2 was very effective. Your final answer of 1/2 arctan x^2 + c is indeed correct and can be considered good enough. However, if you want to tweak it a bit, you can also express it as 1/2 arctan (1+x^2) + c. This is just a different way of writing the same answer and may be considered more simplified. Overall, your work shows a strong understanding of integration techniques and I have no doubt that you will do well in Calculus. Keep up the good work!

## What is an integral?

An integral is a mathematical concept that represents the area under a curve on a graph. It is used to find the total value of a function over a given range.

## How do you solve an integral?

To solve an integral, you need to use a combination of algebraic manipulation and integration techniques. This involves finding an antiderivative of the function and evaluating it over the given range.

## What is the general process for solving integrals?

The general process for solving integrals involves identifying the type of integral (definite or indefinite), finding an antiderivative, applying any necessary integration techniques, and evaluating the integral over the given range.

## What is the purpose of solving integrals?

Solving integrals allows us to find the total value of a function over a given range. This can be useful in many real-world applications, such as calculating areas, volumes, and rates of change.

## What are some common integration techniques used to solve integrals?

Some common integration techniques include substitution, integration by parts, partial fractions, and trigonometric substitution. These techniques can help simplify the integral and make it easier to solve.

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