# Integral ( getting started

• renob
In summary, an integral is a mathematical concept used to calculate the total value of a function over a given interval by representing the area under a curve in a graph. To calculate an integral, the Fundamental Theorem of Calculus can be used, which involves taking the antiderivative of a function and evaluating it at the upper and lower limits of the integral. There are two types of integrals: definite, which have specific limits, and indefinite, which represent the general antiderivative of a function. Integrals have various real-world applications in fields such as physics, engineering, and economics. Common methods for solving integrals include substitution, integration by parts, and using trigonometric identities, but numerical methods may also be used when the integral

## Homework Statement

$$\int(x^4+2x^2+x+1)/(x^2+1)^x$$

## The Attempt at a Solution

am I supposed to multiply out the denominator then do u sub?

nevermind i think its arctan

It isn't anything if the exponent on that denominator is really "x"! If it was supposed to be "2", I recommend dividing the fraction our so that you have a polynomial plus a "proper fraction" of the form $$\frac{ax+ b}{x^2+ 1}$$. That you divide into two parts: $$\frac{ax}{x^2+1}$$ and $$\frac{1}{x^2+ 1}$$. The first uses the simple substitution $$u= x^2+ 1$$ and the second is an arctan.

## 1. What is the definition of an integral?

An integral is a mathematical concept that represents the area under a curve in a graph. It is used to calculate the total value of a function over a given interval.

## 2. How do I calculate an integral?

To calculate an integral, you can use the Fundamental Theorem of Calculus, which involves taking the antiderivative of a function and evaluating it at the upper and lower limits of the integral.

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

A definite integral has specific upper and lower limits, while an indefinite integral does not have any limits and represents the general antiderivative of a function.

## 4. How is an integral used in real-world applications?

Integrals are commonly used in physics, engineering, and economics to solve problems related to areas, volumes, and rates of change. They are also used in calculating probabilities and in optimization problems.

## 5. What are some common methods for solving integrals?

Some common methods for solving integrals include substitution, integration by parts, and using trigonometric identities. Additionally, numerical methods such as the trapezoidal rule or Simpson's rule can be used when the integral cannot be solved analytically.