Project ID:
7500020172
Question:
Create matching triples, each containing a function $f$, an algebraically modified form of $f$, and an antiderivative $F$ of $f$.
Header 1:
Function $f$
Header 2:
Function $f$ after algebraic modification
Header 3:
Antiderivative $F$
Text 11:
$$
f(x)=\frac{1}{\sin^2 x \cos^2 x}
$$
Text 21:
$$
f(x)=\frac{\sin^2 x +\cos^2 x}{\sin^2 x \cos^2 x}
$$
Text 31:
$$
F(x) = \mathrm{tg}\,x - \mathrm{cotg}\,x
$$
Text 12:
$$
f(x)=\frac{\cos 2x}{\sin x +\cos x}
$$
Text 22:
$$
f(x)=\frac{\cos^2 x - \sin^2 x}{\sin x + \cos x}
$$
Text 32:
$$
F(x) = \sin x + \cos x
$$
Text 13:
$$
f(x)=\frac{\cos 2x}{\sin^2 x \cos^2 x}
$$
Text 23:
$$
f(x)=\frac{\cos^2 x - \sin^2 x}{\sin^2 x \cos^2 x}
$$
Text 33:
$$
F(x) = - \mathrm{cotg}\,x - \mathrm{tg}\,x
$$
Text 14:
$$
f(x)=\frac{1+\cos 2x}{2\sin^2 x \cos^2 x}
$$
Text 24:
$$
f(x)=\frac{1+\cos^2 x-\sin^2 x}{2\sin^2 x \cos^2 x}
$$
Text 34:
$$
F(x) = - \mathrm{cotg}\,x
$$
Text 15:
$$
f(x)=\frac{1+ \sin 2x}{\sin x + \cos x}
$$
Text 25:
$$
f(x)=\frac{\sin^2 x+ \cos^2 x+ 2\sin x \cos x}{\sin x + \cos x}
$$
Text 35:
$$
F(x) = - \cos x + \sin x
$$
Text 16:
$$
f(x)=\frac{2 \mathrm{tg}\,x}{\sin 2x }
$$
Text 26:
$$
\frac{2\frac{\sin x}{\cos x}}{2\sin x \cos x}
$$
Text 36:
$$
F(x) = \mathrm{tg}\,x
$$
Workflow:
translation