How to Solve Integrals Example 1

How to Solve Integrals Example 1

Okay so today, we’re going to do a bunch of basic integral problems without any kind of substitution, just merely basic integral so that we can start getting the idea about how to take the antiderivative. So the first thing we’re going to do is; 3x2 + 2x + 1 dx. We’re going to take this antiderivative or integral term by term. The way that we’re going to do that is we look at the x’s first.

So, I'm going to write the x here. I'm going to deal with this term first, the 3x2. So, I write the x. the first thing we do with the antiderivative is add 1 to each of the exponents. So, instead of 2 here, we’re going to add 1 and it's going to be 3 because if you remember when you take the derivative, the exponent always decreases by 1. So when we’re taking the antiderivative, it's going to increase by 1.

So we’re going to have x3 and the second step whenever you're taking the integral is to divide the coefficient by the new exponent. So in this case, 3 the coefficient divided by the new exponent 3, 3/3 is 1. So we could write 1 there but it disappears. And you know that that’s true because if you take the derivative again, add x3, you would get 3x2. So you know that we did it correctly.

So x3 and then with this term now, x, the exponent here is 1. So we add 1 and the new exponent is 2. We take the coefficient and we divide by the new exponent, 2/2 is 1. So we don’t need to write that. And then the antiderivative of 1 which is always just x. The derivative of x would of course be 1. So the antiderivative of 1 is x.

And then whenever we’re getting the antiderivative or the integral of something, we always have to put + c which stands for a constant because in this function here, we can have say + 1 or + 2 or + 3 or whatever constant would be here. And if we took the derivative, that would disappear and you wouldn’t see it in this equation up here. So you never know if there was a constant on the integral function before. You always have to add c to cover that. That’s the answer.


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