# How do you find the integral #int_0^13dx/(root3((1+2x)^2)# ?

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To find the integral ∫₀¹₃ dx/√₃(1 + 2x)², you can use the substitution method. Let u = 1 + 2x. Then, du/dx = 2, which implies dx = du/2.

Substituting these into the integral, we have: ∫₀¹₃ dx/√₃(1 + 2x)² = ∫₀¹₃ (1/2) * du / √₃(u²)

This simplifies to: (1/2√₃) ∫₀¹₃ du/u

Now, integrate with respect to u: (1/2√₃) [ln|u|] from 1 to 7

Substitute back u = 1 + 2x: (1/2√₃) [ln|1 + 2x|] from 1 to 7

Evaluate at the upper and lower limits: (1/2√₃) [ln|1 + 14| - ln|1 + 2|]

Finally, simplify: (1/2√₃) [ln 15 - ln 3]

So, the integral evaluates to: (1/2√₃) [ln(15/3)] = (1/2√₃) ln(5)

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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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