# How do you integrate #int t(2t+7)^(1/3)# by integration by parts method?

We start with:

Integration by parts tells us that:

So the integral will now become:

We can now directly integrate that to finish the task and get:

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To integrate ( \int t(2t+7)^{\frac{1}{3}} ) using integration by parts, let's use the formula:

[ \int u , dv = uv - \int v , du ]

Here, we can let ( u = t ) and ( dv = (2t+7)^{\frac{1}{3}} , dt ). Then we find ( du ) and ( v ) accordingly.

[ du = dt ] [ v = \frac{3}{5}(2t+7)^{\frac{5}{3}} ]

Now, apply the integration by parts formula:

[ \int t(2t+7)^{\frac{1}{3}} , dt = \frac{3}{5}t(2t+7)^{\frac{5}{3}} - \int \frac{3}{5}(2t+7)^{\frac{5}{3}} , dt ]

This is a simpler integral that can be evaluated straightforwardly.

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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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