De onbepaalde integraal

is gelijk aan
|
A. − cos + C |
B. − cos x + C |
C. − cos x + C |
D. − cos x + C |
E. \(\ln \frac {1} {\cos^2\frac x2} \) + C |
[ 6-5577 - op net sinds 15.6.11-(E)-4.11.2023 ]
Translation in E N G L I S H
Evaluate the indefinite integral

|
A. − cos + C |
B. − cos x + C |
C. − cos x + C |
D. − cos x + C |
E. \(\ln \frac {1} {\cos^2\frac x2} \) + C |
Oplossing - Solution
\( \int \frac{\tan \frac x2}{1+\tan^2 \frac x2}dx = \int (\tan \frac x2\cdot \cos^2 \frac x2)\;dx
= \int (\sin \frac x2 \cdot \cos \frac x2) \;dx \\
= \frac12\cdot \int (2\cdot \sin \frac x2 \cdot \cos \frac x2) \;dx =
= \frac12 \int \sin x\;dx = -\,\frac12\,\cos x+C
\)