Today we are going to discuss definite integrals and how to solve them. Definite integrals are a type of integration that gives us an exact value of the area under a curve between two given points. In order to solve them, we need to use a wide range of integration techniques, which we will cover in this post. We also have some great examples that will help explain the concept of definite integrals in a better way.
Example 1: Integration by Substitution
To use integration by substitution, we need to replace the variable in the integral with a new variable. Let's take the example of integrating the function 4x(5x^2+1)^6, from x=0 to x=1. The first step is to set 5x^2+1=u. Then, differentiate both sides of the equation to get 10x dx = du. We can then substitute this into the integral to get ∫ (4/10)u^6 du. Solving this, we get the answer as 139/25005.
Example 2: Integration by Parts
Integration by parts is used to integrate the product of two functions. Let's take the example of integrating the function xe^x, from x=0 to x=1. To use integration by parts, we need to choose one function as u and the other as dv/dx. Assuming u=x and dv/dx=e^x, we can differentiate and integrate to get the solution to the definite integral as e-1.
Conclusion
Definite integrals are an important part of calculus and are used in many fields, including physics, economics, and engineering. By understanding the different techniques used to solve definite integrals, we can easily evaluate the exact value of the area under a curve between two given points. Through example problems, we have learned how to solve two types of definite integrals: integration by substitution and integration by parts. We hope these examples help you understand the topic better.
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