Evaluate an Integral
Step 1:
Enter an expression below to find the indefinite integral, or add bounds to solve for the definite integral. Make sure to specify the variable you wish to integrate with.
Step 2:
Click the blue arrow to compute the integral.
More Information
To get started, try working from the example problem already populated in the box above. Just click the blue arrow and you'll see a solved example. Modify that expression as needed. A few examples of solved integrals are provided below as well. These are the types of problems you can solve with this tool.
Remember, these solvers are great for checking your work, experimenting with different equations, or reminding yourself how to work a particular problem. Maybe you just need a quick answer at work and don't want to solve the problem by hand. But if you're working on homework your teacher is going to want to see how you solved the problem step-by-step to make sure you understand the process. It's not just about getting the right answer (afterall, a computer can do that!) but learning how integrals behave and how they can be used. So, consider going back to our list of calculus lessons to review.
Examples solved with the tool above:
Solve: \( \int_0^5 4xdx \) Solution: \(50\)
Find the indefinite integral: \(\int 4x^2+7\) Solution: \( \frac{4}{3}x^3+7x+C\)
Integrate the sine: \(\int_0^\pi sinx\) Solution: \(2\)
Further Reading
You might be looking for other lessons that cover integrals and integration to help further understand what the tool above is actually doing. After all, it does you no good to simply have the answer unless you know it's right and why it's right!
Integral Questions:
- Difficult integration: int [ 1 / (1 + x^4) ] dx
- Find the Integral of Sqrt(x^2+2x)dx
- Area bounded by y = x^2 and by its normal
- Evaluating the line integral ?c y dx +x^2 dy
More Calculators
- Derivative Calculator
- Factor Calculator
- Equation Solver
Video transcript
- [Instructor] Let's say that we have the function g of x, and it is equal to the definite integral from 19 to x of the cube root of t dt. And what I'm curious about finding or trying to figure out is, what is g prime of 27? What is that equal to? Pause this video and try to think about it, and I'll give you a little bit of a hint. Think about the second fundamental theorem of calculus. All right, now let's work on this together. So we wanna figure out what g prime, we could try to figure out what g prime of x is, and then evaluate that at 27, and the best way that I can think about doing that is by taking the derivative of both sides of this equation. So let's take the derivative of both sides of that equation. So the left-hand side, we'll take the derivative with respect to x of g of x, and the right-hand side, the derivative with respect to x of all of this business. Now, the left-hand side is pretty straight forward. The derivative with respect to x of g of x, that's just going to be g prime of x, but what is the right-hand side going to be equal to? Well, that's where the second fundamental theorem of calculus is useful. I'll write it right over here. Second fundamental, I'll abbreviate a little bit, theorem of calculus. It tells us, let's say we have some function capital F of x, and it's equal to the definite integral from a, sum constant a to x of lowercase f of t dt. The second fundamental theorem of calculus tells us that if our lowercase f, if lowercase f is continuous on the interval from a to x, so I'll write it this way, on the closed interval from a to x, then the derivative of our capital f of x, so capital F prime of x is just going to be equal to our inner function f evaluated at x instead of t is going to become lowercase f of x. Now, I know when you first saw this, you thought that, "Hey, this might be some cryptic thing "that you might not use too often." Well, we're gonna see that it's actually very, very useful and even in the future, and some of you might already know, there's multiple ways to try to think about a definite integral like this, and you'll learn it in the future. But this can be extremely simplifying, especially if you have a hairy definite integral like this, and so this just tells us, hey, look, the derivative with respect to x of all of this business, first we have to check that our inner function, which would be analogous to our lowercase f here, is this continuous on the interval from 19 to x? Well, no matter what x is, this is going to be continuous over that interval, because this is continuous for all x's, and so we meet this first condition or our major condition, and so then we can just say, all right, then the derivative of all of this is just going to be this inner function replacing t with x. So we're going to get the cube root, instead of the cube root of t, you're gonna get the cube root of x. And so we can go back to our original question, what is g prime of 27 going to be equal to? Well, it's going to be equal to the cube root of 27, which is of course equal to three, and we're done.