# How do you find the antiderivative of #(5x^2)/(x^2 + 1)#?

The integral becomes:

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

To find the antiderivative of (\frac{5x^2}{x^2 + 1}), you can use the method of partial fraction decomposition. First, rewrite the fraction as:

[\frac{5x^2}{x^2 + 1} = \frac{Ax + B}{x^2 + 1}]

Then, multiply both sides by (x^2 + 1) to clear the denominator:

[5x^2 = (Ax + B)(x^2 + 1)]

Expand the right side:

[5x^2 = Ax^3 + Ax + Bx^2 + B]

Now, equate coefficients of like terms:

For (x^2) terms: (0x^2 = B) For (x) terms: (5x = Ax) For constant terms: (0 = A)

From the constant term equation, (A = 0), and from the (x) term equation, (5x = Ax) implies (A = 5).

So, the partial fraction decomposition is:

[\frac{5x^2}{x^2 + 1} = \frac{5}{x^2 + 1}]

Now, you can integrate term by term. The integral of (\frac{5}{x^2 + 1}) is (5\tan^{-1}(x) + C), where (C) is the constant of integration.

Therefore, the antiderivative of (\frac{5x^2}{x^2 + 1}) is (5\tan^{-1}(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.

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