# Evaluate #int \ (1+sqrt(x))^9/sqrt(x) \ dx #?

If we consider the integral,

We conclude that,

We can therefore rewrite the given integral easily so it is in this form.

Using this general result we conclude,

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# int \ (1+sqrt(x))^9/sqrt(x) \ dx = (1+sqrt(x))^10/5 + C #

We want to evaluate:

We can perform a simple substitution. Let:

Substituting into the integral we get:

Restoring the substitution we get:

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To evaluate ( \int \frac{(1+\sqrt{x})^9}{\sqrt{x}} , dx ), we can simplify the expression by expanding ( (1+\sqrt{x})^9 ) using the binomial theorem. After expanding, we integrate each term separately. The integral of ( \sqrt{x} ) is ( \frac{2}{3}x^{3/2} + C ).

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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.

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