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人気のある微分積分 >

integral of sin(ln(8x))

解

\int sin (ln (8 x)) d x

解

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

解答ステップ

\int sin (ln (8 x)) d x

u置換積分法を適用する

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

定数を除く:

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

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

部分積分を適用する

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

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

= \frac{1}{8} (u sin (ln (u)) - (\frac{1}{2} u sin (ln (u)) + \frac{1}{2} u cos (ln (u))))

代用を戻す

u = 8 x

= \frac{1}{8} (8 x sin (ln (8 x)) - (\frac{1}{2} \cdot 8 x sin (ln (8 x)) + \frac{1}{2} \cdot 8 x cos (ln (8 x))))

簡素化

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

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

定数を解答に追加する

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

グラフ

人気の例

derivative arcsin(x)

d er i v a t i v e arcsin (x)

(d^2y)/(dx^2)+2/x (dy)/(dx)=3xx

\frac{d ^{2} y}{d x ^{2}} + \frac{2}{x} \frac{d y}{d x} = 3 xx

tangent f(x)=(x^2+7)(5x+4),\at x=2

t an g e n t f (x) = (x^{2} + 7) (5 x + 4), at x = 2

limit as x approaches 0 of 9

x \to 0 lim (9)

integral of 1/(cos^2(3x))

\int \frac{1}{cos ^{2} ( 3 x )} d x