Solving Definite Integrals with Variable Limits

[CalculusHelp1]

Homework Statement

I'm trying to solve a problem where you are asked to find the derivative of an integral where both limits of integrations contain variables

Homework Equations

Definition of integrals, derivatives

The Attempt at a Solution

For problems where you have an integral from a to x, where a is a constant and x is the variable and you are asked to find the derivative of the integral, you can just apply the fundamental theorem of calculus and arrive at the original function f(x).

For problems where you have an integral from a to a function of x, such as x^2, you can use 'u' substitution and find the derivative of the integral from a to u multiplied by du/dx

What can you do for problems where both the lower and upper bounds are variables, e.g. derivative of an integral from x to x^2-x. How would you approach one of these problems?

I can answer questions of the first two kinds but am stuck on the third type.

 

[Mark44]

Homework Statement

I'm trying to solve a problem where you are asked to find the derivative of an integral where both limits of integrations contain variables

Homework Equations

Definition of integrals, derivatives

The Attempt at a Solution

For problems where you have an integral from a to x, where a is a constant and x is the variable and you are asked to find the derivative of the integral, you can just apply the fundamental theorem of calculus and arrive at the original function f(x).

For problems where you have an integral from a to a function of x, such as x^2, you can use 'u' substitution and find the derivative of the integral from a to u multiplied by du/dx

What can you do for problems where both the lower and upper bounds are variables, e.g. derivative of an integral from x to x^2-x. How would you approach one of these problems?

I can answer questions of the first two kinds but am stuck on the third type.

You can split the integral into two integrals. For example,
$$\int_x^{x^2 -x} f(x) dx = \int_x^a f(x) dx + \int_a^{x^2-x} f(x)dx$$
$$= -\int_a^x f(x) dx + \int_a^{x^2-x} f(x)dx$$

 

[CalculusHelp1]

Thanks a lot!

 

[CalculusHelp1]

Sorry one more question, how would you deal with finding the derivative if there was another function being multiplied by the integral?

For example, d(x^5 * integral of f(x) from x to x^2-x)/dx?

Do you use the product rule here or is there some other trick?

 

[CalculusHelp1]

Nevermind I figured it out.

Problems solved.