5 years back

Calculus Level 5

$\large \displaystyle\int \dfrac{\sec^2(x) - 2010}{\sin^{2010}(x)}\, dx = \dfrac{P(x)}{(\sin(x))^{2010}} + C$

If the equation above is true for some function $P(x)$ with $P ( \frac{ \pi}{4} ) = \frac{1}{2^{1005} }$ , where $C$ denotes the arbitrary constant, then find the value of $P\left(\dfrac{\pi}{3}\right)$ .

\dfrac{1}{\sqrt3}

\sqrt3

1 0

3\sqrt3

2 solutions

Tanishq Varshney
Sep 11, 2015

its simple

$\large{\int \frac{\sec^2 x}{(\sin x)^{2010}}dx-\int \frac{2010}{(\sin x)^{2010}}dx}$

In the former integral apply integration by parts considering $\frac{1}{(\sin x)^{2010}}$ as first function and $\sec^2 x$ as second

$\large{\frac{\tan x}{(\sin x)^{2010}}+\int {\frac{2010}{(\sin x)^{2010}}}dx-\int {\frac{2010}{(\sin x)^{2010}}}dx}$

so thus $P(x)=\tan (x)$

Can you explain what the " $\not$ " symbol means in the integration?

Calvin Lin Staff - 5 years, 9 months ago

Oh actually I wanted to show the terms cancel out

Tanishq Varshney - 5 years, 9 months ago

Ah. Unfortunately you will not be able to do a complete strikethrough using Latex (without having additional packages installed in our backend).

I've edited the problem. Can you edit your solution and indicate why that condition is necessary? IE point out what was incomplete in your solution. If you need help, look at Jake's report.

Calvin Lin Staff - 5 years, 9 months ago

Lu Chee Ket
Nov 1, 2015

Just a slight thing to scare people from answering.

Lu Chee Ket - 5 years, 7 months ago

5 years back

2 solutions

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