# What is #f(x) = int xcosx dx# if #f(pi/4)=-2 #?

We have to integrate by using, the Rule of Integration by Parts :

By signing up, you agree to our Terms of Service and Privacy Policy

To find ( f(x) = \int x \cos(x) , dx ) when ( f(\frac{\pi}{4}) = -2 ), we can first differentiate ( f(x) ) using integration by parts:

[ \int u , dv = uv - \int v , du ]

Let ( u = x ) and ( dv = \cos(x) , dx ), then ( du = dx ) and ( v = \sin(x) ).

[ \int x \cos(x) , dx = x\sin(x) - \int \sin(x) , dx ]

[ = x\sin(x) + \cos(x) + C ]

Given that ( f(\frac{\pi}{4}) = -2 ), we can plug this into the equation:

[ -2 = \frac{\pi}{4}\sin\left(\frac{\pi}{4}\right) + \cos\left(\frac{\pi}{4}\right) + C ]

Solve for ( C ):

[ C = -2 - \frac{\pi}{4}\sin\left(\frac{\pi}{4}\right) - \cos\left(\frac{\pi}{4}\right) ]

[ = -2 - \frac{\pi}{4}\left(\frac{\sqrt{2}}{2}\right) - \frac{\sqrt{2}}{2} ]

[ = -2 - \frac{\pi\sqrt{2}}{8} - \frac{\sqrt{2}}{2} ]

[ = -2 - \frac{\pi\sqrt{2} + 4\sqrt{2}}{8} ]

[ = -2 - \frac{\sqrt{2}(\pi + 4)}{8} ]

Therefore, ( f(x) = x\sin(x) + \cos(x) - 2 - \frac{\sqrt{2}(\pi + 4)}{8} ).

By signing up, you agree to our Terms of Service and Privacy Policy

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.

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7