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人気のある微分積分 >

integral of (1+sin(2x))^2

解

\int (1 + sin (2 x))^{2} d x

解

\frac{3}{2} x - \frac{1}{8} sin (4 x) - cos (2 x) + C

解答ステップ

\int (1 + sin (2 x))^{2} d x

u置換積分法を適用する

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

\int \frac{1}{2} d u = \frac{1}{2} u

\int sin (u) d u = - cos (u)

\int \frac{1}{2} sin^{2} (u) d u = \frac{1}{4} (u - \frac{1}{2} sin (2 u))

= \frac{1}{2} u - cos (u) + \frac{1}{4} (u - \frac{1}{2} sin (2 u))

代用を戻す

u = 2 x

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

簡素化

\frac{1}{2} \cdot 2 x - cos (2 x) + \frac{1}{4} (2 x - \frac{1}{2} sin (2 \cdot 2 x)) : \frac{3}{2} x - \frac{1}{8} sin (4 x) - cos (2 x)

= \frac{3}{2} x - \frac{1}{8} sin (4 x) - cos (2 x)

定数を解答に追加する

= \frac{3}{2} x - \frac{1}{8} sin (4 x) - cos (2 x) + C

グラフ

人気の例

\frac{d}{d t} (csc (t))

integral of (sin(x))^3(cos(x))^4

\int (sin (x))^{3} (cos (x))^{4} d x

(2x^3+y)dx-xdy=0

(2 x^{3} + y) d x - x d y = 0

d/(dt)(1+2cos(t))

\frac{d}{d t} (1 + 2 cos (t))

derivative of 1-2cos(x)

\frac{d}{d x} (1 - 2 cos (x))