How Do You Find the Integral of e^sqrt(x)?

TL;DR

To find the integral of e raised to the square root of x, start with u substitution where u equals the square root of x. Then apply integration by parts using the formula ∫u dv = uv - ∫v du. The final result is 2sqrt(x)e^(sqrt(x)) - 2e^(sqrt(x)) + C.

Transcript

in this video we're going to talk about how to find the integral of e raised to the square root X DX so how can we do it well first we need to start with u substitution we're gonna make u equal to the square root of x and the square root of x is basically x raised to the one half so now we'll need to find D u used in a power rule so it's gonna be 1... Read More

Key Insights

• 💄 U substitution is a technique used to simplify integrals by making a substitution for a variable.
• ✊ The power rule is used to find the derivative of a function with a variable raised to a constant power.
• 🥳 Integration by parts is a method that involves breaking down the integral of a product into two parts and using a formula to simplify it.
• 🎓 The formula for integration by parts is integral u * dv = u * v - integral v * du.

Questions & Answers

Q: How do we find the integral of e raised to the square root of X?

To find the integral, start with u substitution by letting u equal the square root of X, then apply integration by parts using the formula integral u * dv = u * v - integral v * du.

Q: What is the next step after isolating dx and replacing variables?

The next step is to apply integration by parts by using the formula integral u * dv = u * v - integral v * du. In this case, u is equal to u and v is equal to e to the u.

Q: How do we simplify the integral expression after applying integration by parts?

After distributing the constant 2 and simplifying the integral expression, it becomes 2u * e to the u - 2 * e to the u + C.

Q: How do we find the final answer for the integral of e to the square root of X DX?

To find the final answer, replace the u variable with the square root of X in the simplified expression, resulting in 2 square root X * e to the square root X - 2e to the square root X + C.

Summary & Key Takeaways

• Start with u substitution by letting u equal the square root of X.

• Use the power rule to find du, which becomes 1/2X^(1/2).

• Multiply both sides of the equation by 2 square root X to isolate dx, resulting in dx on the right side.

• Replace the square root of X with the variable u and dx with 2 square root X du.

• Apply integration by parts by using the formula integral u * dv = u * v - integral v * du.