# Integration: Evaluate the Definite Integral?

• KAISER91
In summary, integration is a mathematical concept used to find the area under a curve or accumulation of a quantity over an interval. It is important for solving real-world problems involving rates of change. There are two types of integration: indefinite and definite. Indefinite integration gives a general antiderivative while definite integration gives a specific value between two points. To evaluate a definite integral, you first find the indefinite integral and then substitute the upper and lower limits of integration. Some common methods for evaluating definite integrals include the Fundamental Theorem of Calculus, substitution, integration by parts, and trigonometric substitution. To check your answer, you can take the derivative of the definite integral or use online calculators and graphing tools.
KAISER91

## Homework Statement

Q) Evaluate The Definite Integral:

∫ (x^3) / (1 + x^4) dx

Upper Limit: 1
Lower Limit: 0

## The Attempt at a Solution

I think I'm on the right track;

u = 1 + x^4
du/dx = 4x^3
du = 4x^3 dx
1/4 du = x^3 dx

When x = 0 ; u = 1
When x = 1 ; u = 2

Therefore;

∫ (1/4 du) / u

1/4 ∫ u^-1

I'm not sure if that last step is correct; but here is where I get stuck.

Help will be appreciated. Thanks.

BTW;

Answer given is (1/4) ln 2

$$\int \frac{dx}{x} = ln(x) + C$$

LMAOO!

Silly me.

Nevermind, I got it now.

## 1. What is integration and why is it important?

Integration is a mathematical concept that involves finding the area under a curve or the accumulation of a quantity over an interval. It is important because it allows us to solve real-world problems involving rates of change, such as calculating the distance traveled by an object with a changing velocity or finding the total cost of a varying rate of production.

## 2. What is the difference between indefinite and definite integration?

Indefinite integration involves finding the most general antiderivative of a function, while definite integration involves finding the specific value of the area under the curve between two given points. In other words, indefinite integration gives a function while definite integration gives a number.

## 3. How do you evaluate a definite integral?

To evaluate a definite integral, you first need to find the indefinite integral of the function. Then, you substitute the upper and lower limits of integration into the indefinite integral and subtract the resulting values to find the definite integral.

## 4. What are some common methods for evaluating definite integrals?

Some common methods for evaluating definite integrals include the Fundamental Theorem of Calculus, substitution, integration by parts, and trigonometric substitution. Which method to use depends on the complexity of the function and your familiarity with the different techniques.

## 5. How can I check if my answer to a definite integral is correct?

You can check your answer by taking the derivative of the definite integral. If the derivative matches the original function, then your answer is correct. You can also use online calculators or graphing tools to visualize the area under the curve and compare it to your calculated answer.

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