Find .

The presence of the term leads us to attempt a trigonometric substitution of the form or . For reasons that will become clear in a moment, we will choose the latter. The limits *x*=-1 and *x*=1 become *θ*=π and *θ*=0, respectively, while , so we have

.

Now, the key is to note the trigonometric identity , so we see

.

Now, since we are integrating from zero to π, is positive for the entire interval, and so we may remove the absolute values:

.

Next, we can use again via *θ*→*θ*/2:

,

and since

then

.

Plugging these in,

,

and since and are both positive for our entire region of integration, we have

,

and since the half-angle formula for tangent says

,

,

and via ,

,

We can find an exact value for the secant above using half-angle identities along with and , we can find that

,

so

.

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Tags: Integral, Math, Monday Math, Trigonometry

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