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人気のある微分積分 >

integral of cos(aln(x/b))

解

\int cos (a ln (\frac{x}{b})) d x

解

\frac{a}{a ^{2} + 1} x sin (a ln (\frac{x}{b})) + \frac{1}{a ^{2} + 1} x cos (a ln (\frac{x}{b})) + C

解答ステップ

\int cos (a ln (\frac{x}{b})) d x

u置換積分法を適用する

= \int \frac{e ^{\frac{u}{a}} b cos ( u )}{a} d u

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{b}{a} \cdot \int e^{\frac{u}{a}} cos (u) d u

u置換積分法を適用する

= \frac{b}{a} \cdot \int e^{v} cos (v a) a d v

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{b}{a} a \cdot \int e^{v} cos (v a) d v

部分積分を適用する

= \frac{b}{a} a (e^{v} \frac{1}{a} sin (v a) - \int e^{v} \frac{1}{a} sin (v a) d v)

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{b}{a} a (e^{v} \frac{1}{a} sin (v a) - \frac{1}{a} \cdot \int e^{v} sin (v a) d v)

部分積分を適用する

= \frac{b}{a} a (e^{v} \frac{1}{a} sin (v a) - \frac{1}{a} (- e^{v} \frac{1}{a} cos (a v) - \int - e^{v} \frac{1}{a} cos (a v) d v))

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{b}{a} a (e^{v} \frac{1}{a} sin (v a) - \frac{1}{a} (- e^{v} \frac{1}{a} cos (a v) - (- \frac{1}{a} \cdot \int e^{v} cos (a v) d v)))

このため

\frac{b}{a} \cdot \int e^{v} cos (a v) d v a = \frac{b}{a} a (e^{v} \frac{1}{a} sin (v a) - \frac{1}{a} (- e^{v} \frac{1}{a} cos (a v) - (- \frac{1}{a} \cdot \int e^{v} cos (a v) d v)))

分離する

\int e^{v} cos (a v) d v

= \frac{b}{a} (\frac{a e ^{v} sin ( a v )}{a ^{2} + 1} + \frac{e ^{v} cos ( a v )}{a ^{2} + 1}) a

代用を戻す

= \frac{b}{a} \frac{a e ^{\frac{a l n ( \frac{x}{b} )}{a}} sin ( a \frac{a l n ( \frac{x}{b} )}{a} )}{a ^{2} + 1} + \frac{e ^{\frac{a l n ( \frac{x}{b} )}{a}} cos ( a \frac{a l n ( \frac{x}{b} )}{a} )}{a ^{2} + 1} a

簡素化

\frac{b}{a} \frac{a e ^{\frac{a l n ( \frac{x}{b} )}{a}} sin ( a \frac{a l n ( \frac{x}{b} )}{a} )}{a ^{2} + 1} + \frac{e ^{\frac{a l n ( \frac{x}{b} )}{a}} cos ( a \frac{a l n ( \frac{x}{b} )}{a} )}{a ^{2} + 1} a : \frac{a}{a ^{2} + 1} x sin (a ln (\frac{x}{b})) + \frac{1}{a ^{2} + 1} x cos (a ln (\frac{x}{b}))

= \frac{a}{a ^{2} + 1} x sin (a ln (\frac{x}{b})) + \frac{1}{a ^{2} + 1} x cos (a ln (\frac{x}{b}))

定数を解答に追加する

= \frac{a}{a ^{2} + 1} x sin (a ln (\frac{x}{b})) + \frac{1}{a ^{2} + 1} x cos (a ln (\frac{x}{b})) + C

グラフ

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