# How do I evaluate the indefinite integral #intx*sin(x)*tan(x)dx# ?

Integration by parts can be used to evaluate this integral.

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To evaluate the indefinite integral ∫x*sin(x)*tan(x)dx, you can use integration by parts. Let u = x and dv = sin(x)*tan(x)dx. Then, differentiate u to get du/dx = 1 and integrate dv to get v = -ln|cos(x)|.

Apply the integration by parts formula: ∫u dv = uv - ∫v du.

This yields:
∫x*sin(x)*tan(x)dx = -x*ln|cos(x)| - ∫(-ln|cos(x)|)dx.

The integral of -ln|cos(x)| can be evaluated separately. Let w = -ln|cos(x)| and dz = dx. Then, differentiate w to get dw/dx = -tan(x) and integrate dz to get z = x.

Apply integration by parts again:
∫(-ln|cos(x)|)dx = -x*ln|cos(x)| - ∫x*(-tan(x))dx.

This gives:
∫x*sin(x) tan(x)dx = -xln|cos(x)| + ∫x*tan(x)dx.

Now, you need to evaluate ∫x*tan(x)dx. This integral can be solved using substitution. Let t = cos(x), then dt = -sin(x)dx.

Substituting t = cos(x) and dt = -sin(x)dx into the integral, you get: ∫x*tan(x)dx = ∫(-x)*dt.

Integrating with respect to t gives:
∫x*tan(x)dx = -x*t + ∫tdt.

This simplifies to:
∫x*tan(x)dx = -x*cos(x) + ∫cos(x)dt.

Finally, integrating ∫cos(x)dt gives:
∫x*tan(x)dx = -x*cos(x) + sin(x) + C,

where C is the constant of integration. Therefore, the solution to the indefinite integral is:
∫x*sin(x) tan(x)dx = -xln|cos(x)| - x*cos(x) + 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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