# How do you calculate the anti-derivative of #secx(secx + tanx)dx#?

This integral becomes:

Remembering that:

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To calculate the antiderivative of ( \sec(x)(\sec(x) + \tan(x)) ) with respect to ( x ), you can use the substitution method. Let ( u = \sec(x) + \tan(x) ). Then, ( du = (\sec(x)\tan(x) + \sec^2(x)) , dx ). Rearrange this expression to solve for ( dx ), which gives ( dx = \frac{du}{\sec(x)\tan(x) + \sec^2(x)} ). Now substitute ( u ) and ( dx ) into the integral. This gives:

[ \int \sec(x)(\sec(x) + \tan(x)) , dx = \int u , du ]

Now integrate ( u ) with respect to ( u ), which gives:

[ \int u , du = \frac{u^2}{2} + C ]

Finally, substitute back ( u = \sec(x) + \tan(x) ) into the expression:

[ \int \sec(x)(\sec(x) + \tan(x)) , dx = \frac{(\sec(x) + \tan(x))^2}{2} + C ]

Where ( C ) is the constant of integration.

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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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