# How do you find the indefinite integral of #int (4sinx)/(3tanx)dx#?

The answer is

So,

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To find the indefinite integral of ( \int \frac{4\sin(x)}{3\tan(x)} , dx ), we can use the trigonometric identity ( \tan(x) = \frac{\sin(x)}{\cos(x)} ) to rewrite ( \frac{4\sin(x)}{3\tan(x)} ) as ( \frac{4\sin(x)}{3\left(\frac{\sin(x)}{\cos(x)}\right)} ). Then, we simplify by multiplying the numerator and denominator by ( \cos(x) ) to get ( \frac{4\sin(x)\cos(x)}{3\sin(x)} ). Now, ( \sin(x) ) cancels out, leaving ( \frac{4\cos(x)}{3} ). Thus, the indefinite integral becomes ( \int \frac{4\cos(x)}{3} , dx ), which equals ( \frac{4}{3} \int \cos(x) , dx ). Integrating ( \cos(x) ) yields ( \sin(x) + C ), where ( C ) is the constant of integration. Therefore, the indefinite integral of ( \int \frac{4\sin(x)}{3\tan(x)} , dx ) is ( \frac{4}{3}\sin(x) + C ).

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To find the indefinite integral of ( \int \frac{4 \sin(x)}{3 \tan(x)} , dx ), we can rewrite the expression in terms of sine and cosine functions:

[ \tan(x) = \frac{\sin(x)}{\cos(x)} ]

Substituting ( \tan(x) ) with ( \frac{\sin(x)}{\cos(x)} ), we get:

[ \frac{4 \sin(x)}{3 \tan(x)} = \frac{4 \sin(x)}{3 \left(\frac{\sin(x)}{\cos(x)}\right)} = \frac{4 \sin(x) \cos(x)}{3 \sin(x)} = \frac{4 \cos(x)}{3} ]

Now, the integral becomes:

[ \int \frac{4 \cos(x)}{3} , dx ]

This is a straightforward integral:

[ \int \frac{4 \cos(x)}{3} , dx = \frac{4}{3} \int \cos(x) , dx = \frac{4}{3} \sin(x) + C ]

Where ( C ) is the constant of integration.

Therefore, the indefinite integral of ( \int \frac{4 \sin(x)}{3 \tan(x)} , dx ) is ( \frac{4}{3} \sin(x) + 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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