# How do you find the antiderivative of #f(x)=(2x^2-5x-3)/(x-3)#?

By signing up, you agree to our Terms of Service and Privacy Policy

To find the antiderivative of ( f(x) = \frac{{2x^2 - 5x - 3}}{{x - 3}} ), we can use polynomial long division to divide ( 2x^2 - 5x - 3 ) by ( x - 3 ) to rewrite the function in the form ( f(x) = q(x) + \frac{{r(x)}}{{x - 3}} ), where ( q(x) ) is the quotient and ( r(x) ) is the remainder. Once we have ( q(x) ), we integrate it term by term to find the antiderivative.

First, let's perform the polynomial long division:

```
2x + 1
___________________
x - 3 | 2x^2 - 5x - 3
- (2x^2 - 6x)
___________
x - 3
- (x - 3)
__________
0
```

So, ( 2x^2 - 5x - 3 = (x - 3)(2x + 1) + 0 ).

Thus, ( f(x) = 2x + 1 ).

Now, we integrate ( f(x) = 2x + 1 ) term by term:

( \int (2x + 1) , dx = \int 2x , dx + \int 1 , dx )

( = x^2 + x + C ), where ( C ) is the constant of integration.

Therefore, the antiderivative of ( f(x) = \frac{{2x^2 - 5x - 3}}{{x - 3}} ) is ( x^2 + x + C ).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you evaluate the integral #int 1/(x(lnx)^2)dx# from #e# to #oo#?
- What is #int_3^oo 3/x -2/(x-2)dx#?
- How to calculate the integral of 2-sinx/2+sinx?
- If f(3)=13, f' is continuous, and the integral from 3 to 5 f'(x)dx=24 then the value of f(5) is ?
- How do you evaluate the definite integral by the limit definition given #int x^3dx# from [-1,1]?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7