Integral of (x-1)/sqrt(x) from 1 to 9

TL;DR

The video explains how to integrate a complex expression using fraction splitting and the power rule.

Transcript

okay let's do more Integrations the integral from 1 to 9 and here we have a crazy expression x - one on the top over square root of x luckily we only have one thing on the bottom and this is a power function so we should try to split the fraction to see what we can do um after that and here we are going to have one to n let me split the fraction as... Read More

Key Insights

• 😑 The complex expression is split into two fractions before integrating.
• 😑 The power rule is applied to each fraction to simplify the expression.
• 🔌 The values of 1 and 9 are plugged in to evaluate the integral.

Questions & Answers

Q: How is the complex expression in the video split into fractions?

The complex expression is split into two fractions: x/sqrt(x) and -1/(sqrt(x)). For the first fraction, x/sqrt(x), the exponent of 1 on the top is subtracted from the exponent of 1/2 on the bottom to get x^(1/2).

Q: What is the result of integrating the first fraction?

The integral of x^(1/2) is (2/3)x^(3/2).

Q: How is the second fraction simplified?

The second fraction, -1/(sqrt(x)), is brought up with a negative exponent, resulting in -x^(-1/2).

Q: What is the integral of the second fraction?

The integral of -x^(-1/2) is -2x^(1/2).

Summary & Key Takeaways

• The video demonstrates the process of integrating a complex expression from 1 to 9.

• The expression is split into two fractions, and each fraction is simplified using the power rule.

• The final integral is solved and evaluated for the values of 1 and 9.