ja

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

人気のある微分積分 >

integral of sin^5(x)cos^5(x)

解

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

解

- \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{4} - \frac{cos ^{10} ( x )}{10} + C

解答ステップ

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

三角関数の公式を使用して書き換える

= \int (1 - cos^{2} (x))^{2} sin (x) cos^{5} (x) d x

u置換積分法を適用する

= \int - u^{5} (1 - u^{2})^{2} d u

拡張

- u^{5} (1 - u^{2})^{2} : - u^{5} + 2 u^{7} - u^{9}

= \int - u^{5} + 2 u^{7} - u^{9} 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 u^{5} d u + \int 2 u^{7} d u - \int u^{9} d u

\int u^{5} d u = \frac{u ^{6}}{6}

\int 2 u^{7} d u = \frac{u ^{8}}{4}

\int u^{9} d u = \frac{u ^{10}}{10}

= - \frac{u ^{6}}{6} + \frac{u ^{8}}{4} - \frac{u ^{10}}{10}

代用を戻す

u = cos (x)

= - \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{4} - \frac{cos ^{10} ( x )}{10}

定数を解答に追加する

= - \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{4} - \frac{cos ^{10} ( x )}{10} + C

グラフ

人気の例

limit as x approaches-9-of sqrt(x^2-81)

x \to - 9 - lim (x^{2} - 81)

integral of 2x^2cos(mx)

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

limit as x approaches-3 of 4x^2+2x-1

x \to - 3 lim (4 x^{2} + 2 x - 1)

integral of sin^6(pix)

\int sin^{6} (π x) d x

sum from n=7 to infinity of 1/(6^n)

n = 7 \sum \infty \frac{1}{6 ^{n}}