integral of xe^{-inx}

解答

\int x e^{- in x} d x

\frac{1}{n ^{2}} (cos (n x) + n x sin (n x)) + i \frac{1}{n ^{2}} (- sin (n x) + n x cos (n x)) + C

求解步骤

\int x e^{- in x} d x

化简

x e^{- in x} : x cos (n x) - i x sin (n x)

= \int x cos (n x) - i x sin (n x) d x

使用积分加法定则:

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

= \int x cos (n x) d x - \int i x sin (n x) d x

\int x cos (n x) d x = \frac{1}{n ^{2}} (n x sin (n x) + cos (n x))

\int i x sin (n x) d x = i \frac{1}{n ^{2}} (sin (n x) - n x cos (n x))

= \frac{1}{n ^{2}} (n x sin (n x) + cos (n x)) - i \frac{1}{n ^{2}} (sin (n x) - n x cos (n x))

化简

\frac{1}{n ^{2}} (n x sin (n x) + cos (n x)) - i \frac{1}{n ^{2}} (sin (n x) - n x cos (n x)) : \frac{1}{n ^{2}} (cos (n x) + n x sin (n x)) + i \frac{1}{n ^{2}} (- sin (n x) + n x cos (n x))

= \frac{1}{n ^{2}} (cos (n x) + n x sin (n x)) + i \frac{1}{n ^{2}} (- sin (n x) + n x cos (n x))

解答补常数

= \frac{1}{n ^{2}} (cos (n x) + n x sin (n x)) + i \frac{1}{n ^{2}} (- sin (n x) + n x cos (n x)) + C

\int \frac{sec ( x ) tan ( x )}{9 + 4 sec ^{2} ( x )} d x

\int cos (t) e^{t} d t

\frac{d x}{d y} = y^{2} x - x + y^{2} - 1

\int tan^{2} (x) sec^{3} (d) x d x

ma c l a u r in f (x) = arctan (x)