3 years ago

What is the volume of this sphere?

Mathematics

1 answer:

Rashid [163]3 years ago

6 0

Answer:

I need to know the radius or diameter.

Step-by-step explanation:

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How do i find the antiderivative of sin^2(x, cos^2(x and tan^2(x? i know how to antidifferentiate sinx, cosx, and tanx, but thes

Lena [83]

For $\sin^2x$ and $\cos^2x$ , it's helpful to remember the half-angle identities:

$\sin^2x=\dfrac{1-\cos2x}2$
$\cos^2x=\dfrac{1+\cos2x}2$

So

$\displaystyle\int\sin^2x\,\mathrm dx=\frac12\int(1-\cos2x)\,\mathrm dx=\frac x2-\frac{\sin2x}4=C$
$\displaystyle\int\cos^2x\,\mathrm dx=\frac12\int(1+\cos2x)\,\mathrm dx=\frac x2+\frac{\sin2x}4=C$

For $\tan^2x$ , the Pythagorean identity suffices:

$\sin^2x+\cos^2x=1\implies \tan^2x+1=\sec^2x$

which means

$\displaystyle\int\tan^2x\,\mathrm dx=\int(\sec^2x-1)\,\mathrm dx=\tan x-x+C$

since $\dfrac{\mathrm d}{\mathrm dx}\tan x=\sec^2x$ .