# How do you integrate #int x/sqrt(144+x^2)dx# using trigonometric substitution?

Refer below for explanation.

Step 1: Draw It!

The first thing to do with trig substitution problems, especially if you have time, is to draw them out. Note that the expression in the denominator -

Step 2: Define a Few Things

From the image, we see that

Step 3: Trigonometric Substitution

Now we can finally take this information and apply it to the problem. Making substitutions, our integral becomes:

Step 4: Simplification

This tends to be pretty long, so bear with me here.

Remember the Pythagorean Identities from trig? One of those identities is

Hm...think about it for a moment. What function's derivative is

You might think we're done, but wait a minute. The problem was given to us in terms of

Step 5: Reverse Substitution

Wonder why I told you to draw the problem? Because now we can clearly see from the triangle above that

Final answer:

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To integrate (\int \frac{x}{\sqrt{144 + x^2}} , dx) using trigonometric substitution, you can use the substitution (x = 12 \sin(\theta)). Then (dx = 12 \cos(\theta) , d\theta).

Substituting (x) and (dx) into the integral, you get:

[ \int \frac{12 \sin(\theta)}{\sqrt{144 + (12\sin(\theta))^2}} \cdot 12 \cos(\theta) , d\theta ]

Simplify this expression to:

[ \int \frac{12^2 \sin(\theta) \cos(\theta)}{\sqrt{144(1 + \sin^2(\theta))}} , d\theta ]

Now, use the trigonometric identity (\sin^2(\theta) + \cos^2(\theta) = 1) to rewrite the expression inside the square root as (\sqrt{144 \cos^2(\theta)}), which simplifies to (12 \cos(\theta)). The integral becomes:

[ \int \frac{12^2 \sin(\theta) \cos(\theta)}{12 \cos(\theta)} , d\theta ]

Simplify further to:

[ \int 12 \sin(\theta) , d\theta ]

Now, integrate (\int 12 \sin(\theta) , d\theta) to get:

[ -12 \cos(\theta) + C ]

Finally, substitute back (x = 12 \sin(\theta)) to get the final answer:

[ -12 \frac{x}{\sqrt{144 + x^2}} + C ]

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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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