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人気のある微分積分 >

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

解

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

解

\frac{1}{16} (x^{2} - \frac{1}{8} sin (8 x^{2})) + C

解答ステップ

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

u置換積分法を適用する

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

三角関数の公式を使用して書き換える

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

\int 1 d u = u

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

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

代用を戻す

u = x^{2}

= \frac{1}{2} \cdot \frac{1}{8} (x^{2} - \frac{1}{8} sin (8 x^{2}))

簡素化

\frac{1}{2} \cdot \frac{1}{8} (x^{2} - \frac{1}{8} sin (8 x^{2})) : \frac{1}{16} (x^{2} - \frac{1}{8} sin (8 x^{2}))

= \frac{1}{16} (x^{2} - \frac{1}{8} sin (8 x^{2}))

定数を解答に追加する

= \frac{1}{16} (x^{2} - \frac{1}{8} sin (8 x^{2})) + C

グラフ

人気の例

derivative (x^2+1)/(x^2-1)

d er i v a t i v e \frac{x ^{2} + 1}{x ^{2} - 1}

tangent y=x^2+5,(4,21)

t an g e n t y = x^{2} + 5, (4, 21)

limit as u approaches 0 of ln((u+1)/u)

u \to 0 lim (ln (\frac{u + 1}{u}))

normal y=-x^3+2x^2-4,\at x=-1

n or ma l y = - x^{3} + 2 x^{2} - 4, at x = - 1

勾配 (-4.3)(5.3)

s l o p e (- 4.3) (5.3)