Integrals of goniometric functions
Task number: 3114
Convert the following integrals to integrals of a rational function and think about which partial fractions you get (which denominators). You no longer have to calculate the decomposition into partial fractions and the resulting primitive function.
Variant 1
\( \int\frac{1}{\sin x \cos x} \, dx \)
Variant 2
\( \int \frac{1}{\sin x} \, dx \)
Variant 3
\( \int\frac{1}{\cos x \sin^3 x} \, dx \)
Variant 4
\( \int \operatorname{tan}^5 x \, dx \)
Variant 5
\( \int \frac{\cos^4 x + \sin^4 x}{\cos^2 x - \sin^2 x} \, dx \)
Variant 6
\( \int \frac{\sin x}{1 + \sin x} \, dx \)
Variant 7
\( \int \frac{1}{2 \sin x -\cos x + 5} \, dx \)
Variant 8
\( \int \frac{\sin x \cos x}{1 + \sin^4 x} \, dx \)
