integral from-1 to 0 of (x+1)cos(npix)

Lösung

\int_{- 1}^{0} (x + 1) cos (nπ x) d x

\frac{1 - ( - 1 ) ^{n}}{π ^{2} n ^{2}}

Schritte zur Lösung

\int_{- 1}^{0} (x + 1) cos (nπ x) d x

Multipliziere aus

(x + 1) cos (nπ x) : x cos (πn x) + cos (πn x)

= \int_{- 1}^{0} x cos (πn x) + cos (πn x) d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{- 1}^{0} x cos (πn x) d x + \int_{- 1}^{0} cos (πn x) d x

\int_{- 1}^{0} x cos (πn x) d x = \frac{1 - ( - 1 ) ^{n}}{π ^{2} n ^{2}}

\int_{- 1}^{0} cos (πn x) d x = 0

= \frac{1 - ( - 1 ) ^{n}}{π ^{2} n ^{2}} + 0

Vereinfache

= \frac{1 - ( - 1 ) ^{n}}{π ^{2} n ^{2}}

\int_{1}^{3} (x^{2} - x) d x

\int_{- \frac{1}{8}}^{\frac{1}{8}} (sin (2 π x)) d x

\int_{- 6}^{1} - x^{2} - 3 x d x

\int_{0}^{60} \frac{x}{2} d x

\int_{- 1}^{1} (1 - x^{2} + 1) d x