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人気のある微分積分 >

integral of (cos(x)ln(cos(x)))/(csc(x))

解

\int \frac{cos ( x ) ln ( cos ( x ) )}{csc ( x )} d x

解

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

解答ステップ

\int \frac{cos ( x ) ln ( cos ( x ) )}{csc ( x )} d x

u置換積分法を適用する

= \int - u ln (u) d u

定数を除く:

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

= - \int u ln (u) d u

部分積分を適用する

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

\int \frac{u}{2} d u = \frac{u ^{2}}{4}

= - (\frac{1}{2} u^{2} ln (u) - \frac{u ^{2}}{4})

代用を戻す

u = cos (x)

= - (\frac{1}{2} cos^{2} (x) ln (cos (x)) - \frac{cos ^{2} ( x )}{4})

簡素化

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

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

定数を解答に追加する

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

グラフ

人気の例

integral of 1/(sqrt(-x^2-6x+16))

\int \frac{1}{- x ^{2} - 6 x + 16} d x

integral of 2x(x^2+10)^{30}

\int 2 x (x^{2} + 10)^{30} d x

integral of ((9y+14))/((y-2)(y^2+4))

\int \frac{( 9 y + 14 )}{( y - 2 ) ( y ^{2} + 4 )} d y

integral of (sinh(x))/(1+cosh^2(x))

\int \frac{sinh ( x )}{1 + cosh ^{2} ( x )} d x

integral of (tan(4x)-sec(4x))^2

\int (tan (4 x) - sec (4 x))^{2} d x