# How do you integrate #int sec^2sqrtx# by integration by parts method?

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How do you integrate #int sec^2sqrtxdx# by integration by parts method?

How do you integrate

The Rule of Integration by Parts (IbP) states :

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To integrate (\int \sec^2(\sqrt{x}) , dx) using integration by parts, let (u = \sqrt{x}) and (dv = \sec^2(\sqrt{x}) , dx). Then, (du = \frac{1}{2\sqrt{x}} , dx) and (v = \tan(\sqrt{x})).

Applying the integration by parts formula: [ \int u , dv = uv - \int v , du ]

we get: [ \int \sec^2(\sqrt{x}) , dx = \sqrt{x}\tan(\sqrt{x}) - \int \tan(\sqrt{x}) \frac{1}{2\sqrt{x}} , dx ]

Now, (\int \tan(\sqrt{x}) \frac{1}{2\sqrt{x}} , dx) can be integrated using substitution method. Let (t = \sqrt{x}), then (dt = \frac{1}{2\sqrt{x}} , dx).

Substituting (t = \sqrt{x}) into the integral, we have: [ \int \tan(\sqrt{x}) \frac{1}{2\sqrt{x}} , dx = \int \tan(t) , dt ]

This integral can be evaluated directly to get the final result.

So, the integration of (\int \sec^2(\sqrt{x}) , dx) using integration by parts involves a subsequent integral of (\tan(t) , dt), where (t = \sqrt{x}).

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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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