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人気のある微分積分 >

integral of (e^x-cos(x))^2

解

\int (e^{x} - cos (x))^{2} d x

解

\frac{1}{2} e^{2 x} - e^{x} sin (x) - e^{x} cos (x) + \frac{1}{2} x + \frac{1}{4} sin (2 x) + C

解答ステップ

\int (e^{x} - cos (x))^{2} d x

u置換積分法を適用する

= \int \frac{( u - cos ( ln ( u ) ) ) ^{2}}{u} d u

拡張

\frac{( u - cos ( ln ( u ) ) ) ^{2}}{u} : u - 2 cos (ln (u)) + \frac{cos ^{2} ( ln ( u ) )}{u}

= \int u - 2 cos (ln (u)) + \frac{cos ^{2} ( ln ( u ) )}{u} d u

総和規則を適用する:

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

= \int u d u - \int 2 cos (ln (u)) d u + \int \frac{cos ^{2} ( ln ( u ) )}{u} d u

\int u d u = \frac{u ^{2}}{2}

\int 2 cos (ln (u)) d u = u sin (ln (u)) + u cos (ln (u))

\int \frac{cos ^{2} ( ln ( u ) )}{u} d u = \frac{1}{2} (ln (u) + \frac{1}{2} sin (2 ln (u)))

= \frac{u ^{2}}{2} - (u sin (ln (u)) + u cos (ln (u))) + \frac{1}{2} (ln (u) + \frac{1}{2} sin (2 ln (u)))

代用を戻す

u = e^{x}

= \frac{( e ^{x} ) ^{2}}{2} - (e^{x} sin (ln (e^{x})) + e^{x} cos (ln (e^{x}))) + \frac{1}{2} (ln (e^{x}) + \frac{1}{2} sin (2 ln (e^{x})))

簡素化

\frac{( e ^{x} ) ^{2}}{2} - (e^{x} sin (ln (e^{x})) + e^{x} cos (ln (e^{x}))) + \frac{1}{2} (ln (e^{x}) + \frac{1}{2} sin (2 ln (e^{x}))) : \frac{1}{2} e^{2 x} - e^{x} sin (x) - e^{x} cos (x) + \frac{1}{2} x + \frac{1}{4} sin (2 x)

= \frac{1}{2} e^{2 x} - e^{x} sin (x) - e^{x} cos (x) + \frac{1}{2} x + \frac{1}{4} sin (2 x)

定数を解答に追加する

= \frac{1}{2} e^{2 x} - e^{x} sin (x) - e^{x} cos (x) + \frac{1}{2} x + \frac{1}{4} sin (2 x) + C

グラフ

人気の例

integral of 2/(100-x)

\int \frac{2}{100 - x} d x

integral of (x^3+2x^2)/x

\int \frac{x ^{3} + 2 x ^{2}}{x} d x

integral of (x-2)\sqrt[3]{x}

\int (x - 2) 3 x d x

integral of cos(5-4x)

\int cos (5 - 4 x) d x

integral of (x^3)/(1+sqrt(1+x^4))

\int \frac{x ^{3}}{1 + 1 + x ^{4}} d x