integral of x^2sec^2(x)

Lösung

\int x^{2} sec^{2} (x) d x

x^{2} tan (x) - i Li_{2} (- e^{2 i x}) - i x^{2} + 2 x ln (1 + cos (2 x) + i sin (2 x)) + C

Schritte zur Lösung

\int x^{2} sec^{2} (x) d x

Vereinfache

x^{2} : xx

= \int xx sec^{2} (x) d x

Wende die partielle Integration an

= x^{2} tan (x) - \int 2 x tan (x) d x

\int 2 x tan (x) d x = i Li_{2} (- e^{2 i x}) + i x^{2} - 2 x ln (1 + cos (2 x) + i sin (2 x))

= x^{2} tan (x) - (i Li_{2} (- e^{2 i x}) + i x^{2} - 2 x ln (1 + cos (2 x) + i sin (2 x)))

Vereinfache

= x^{2} tan (x) - i Li_{2} (- e^{2 i x}) - i x^{2} + 2 x ln (1 + cos (2 x) + i sin (2 x))

Füge eine Konstante zur Lösung hinzu

= x^{2} tan (x) - i Li_{2} (- e^{2 i x}) - i x^{2} + 2 x ln (1 + cos (2 x) + i sin (2 x)) + C

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