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人気のある微分積分 >

integral of xsin(5x^2)cos(8x^2)

解

\int x sin (5 x^{2}) cos (8 x^{2}) d x

解

\frac{1}{4} (- \frac{1}{13} cos (13 x^{2}) + \frac{1}{3} cos (3 x^{2})) + C

解答ステップ

\int x sin (5 x^{2}) cos (8 x^{2}) d x

u置換積分法を適用する

= \int \frac{sin ( 5 u ) cos ( 8 u )}{2} d u

定数を除く:

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

= \frac{1}{2} \cdot \int sin (5 u) cos (8 u) d u

次の恒等を使用する:

cos (t) sin (s) = \frac{sin ( s + t ) + sin ( s - t )}{2}

= \frac{1}{2} \cdot \int \frac{sin ( 5 u + 8 u ) + sin ( 5 u - 8 u )}{2} d u

定数を除く:

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

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int sin (5 u + 8 u) + sin (5 u - 8 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

= \frac{1}{2} \cdot \frac{1}{2} (\int sin (5 u + 8 u) d u + \int sin (5 u - 8 u) d u)

\int sin (5 u + 8 u) d u = - \frac{1}{13} cos (13 u)

\int sin (5 u - 8 u) d u = \frac{1}{3} cos (3 u)

= \frac{1}{2} \cdot \frac{1}{2} (- \frac{1}{13} cos (13 u) + \frac{1}{3} cos (3 u))

代用を戻す

u = x^{2}

= \frac{1}{2} \cdot \frac{1}{2} (- \frac{1}{13} cos (13 x^{2}) + \frac{1}{3} cos (3 x^{2}))

簡素化

\frac{1}{2} \cdot \frac{1}{2} (- \frac{1}{13} cos (13 x^{2}) + \frac{1}{3} cos (3 x^{2})) : \frac{1}{4} (- \frac{1}{13} cos (13 x^{2}) + \frac{1}{3} cos (3 x^{2}))

= \frac{1}{4} (- \frac{1}{13} cos (13 x^{2}) + \frac{1}{3} cos (3 x^{2}))

定数を解答に追加する

= \frac{1}{4} (- \frac{1}{13} cos (13 x^{2}) + \frac{1}{3} cos (3 x^{2})) + C

グラフ

人気の例

tangent f(x)=5-4/x ,(-1,1)

t an g e n t f (x) = 5 - \frac{4}{x}, (- 1, 1)

integral of x^9e^{x^{10}-9}

\int x^{9} e^{x^{10} - 9} d x

integral of 2e^{2x+5}

\int 2 e^{2 x + 5} d x

sum from n=1 to infinity of arctan(3n)

n = 1 \sum \infty arctan (3 n)

integral of-x+2x^3

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