Integration of Exponential Functions

This video tutorial explains how to find the integral of e raised to the square root of x using u-substitution and integration by parts. The process involves substituting u for the square root of x, finding du, and then applying integration by parts to solve the integral. The final solution is derived by replacing the u variable back with the square root of x.

What is the first step in finding the integral of e raised to the square root of x?

• Perform u-substitution

• Differentiate the function

• Apply the power rule

• Use integration by parts

In the u-substitution method, what is u set equal to?

• x

• e^x

• x^2

• sqrt(x)

How is du expressed in terms of x?

• 2 * x^(1/2) dx

• x^(1/2) dx

• 1/2 * x^(-1/2) dx

• e^x dx

What substitution is made for dx in terms of du?

• dx = x du

• dx = 1/2 * sqrt(x) du

• dx = e^u du

• dx = 2 * sqrt(x) du

What technique is used to solve the integral after substitution?

• Trigonometric substitution

• Integration by parts

• Direct integration

• Partial fraction decomposition

In integration by parts, what is the formula used?

• Integral of u dv = u v / integral of v du

• Integral of u dv = u v * integral of v du

• Integral of u dv = u v + integral of v du

• Integral of u dv = u v - integral of v du

What is the integral of e^u du?

• 1/2 e^u + C

• u e^u + C

• e^u + C

• 2 e^u + C

What is the final expression for the integral of e raised to the square root of x?

• e^(sqrt(x)) + 2 sqrt(x) e^(sqrt(x)) + C

• 2 sqrt(x) e^(sqrt(x)) - 2 e^(sqrt(x)) + C

• sqrt(x) e^(sqrt(x)) + 2 e^(sqrt(x)) + C

• 2 e^(sqrt(x)) - sqrt(x) e^(sqrt(x)) + C

What is the purpose of replacing the u variable with the square root of x in the final step?

• To revert to the original variable

• To simplify the expression

• To differentiate the result

• To check the solution

What constant is added to the final integral expression?

• 0

• 1

• 2C

• C