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 sec^{14}(θ)tan(θ)

解

\int sec^{14} (θ) tan (θ) d θ

解

\frac{tan ^{2} ( θ )}{2} + \frac{3 tan ^{4} ( θ )}{2} + \frac{5 tan ^{6} ( θ )}{2} + \frac{5 tan ^{8} ( θ )}{2} + \frac{3 tan ^{10} ( θ )}{2} + \frac{tan ^{12} ( θ )}{2} + \frac{tan ^{14} ( θ )}{14} + C

解答ステップ

\int sec^{14} (θ) tan (θ) d θ

三角関数の公式を使用して書き換える

= \int sec^{2} (θ) (1 + tan^{2} (θ))^{6} tan (θ) d θ

u置換積分法を適用する

= \int u (1 + u^{2})^{6} d u

拡張

u (1 + u^{2})^{6} : u + 6 u^{3} + 15 u^{5} + 20 u^{7} + 15 u^{9} + 6 u^{11} + u^{13}

= \int u + 6 u^{3} + 15 u^{5} + 20 u^{7} + 15 u^{9} + 6 u^{11} + u^{13} 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 d u + \int 6 u^{3} d u + \int 15 u^{5} d u + \int 20 u^{7} d u + \int 15 u^{9} d u + \int 6 u^{11} d u + \int u^{13} d u

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

\int 6 u^{3} d u = \frac{3 u ^{4}}{2}

\int 15 u^{5} d u = \frac{5 u ^{6}}{2}

\int 20 u^{7} d u = \frac{5 u ^{8}}{2}

\int 15 u^{9} d u = \frac{3 u ^{10}}{2}

\int 6 u^{11} d u = \frac{u ^{12}}{2}

\int u^{13} d u = \frac{u ^{14}}{14}

= \frac{u ^{2}}{2} + \frac{3 u ^{4}}{2} + \frac{5 u ^{6}}{2} + \frac{5 u ^{8}}{2} + \frac{3 u ^{10}}{2} + \frac{u ^{12}}{2} + \frac{u ^{14}}{14}

代用を戻す

u = tan (θ)

= \frac{tan ^{2} ( θ )}{2} + \frac{3 tan ^{4} ( θ )}{2} + \frac{5 tan ^{6} ( θ )}{2} + \frac{5 tan ^{8} ( θ )}{2} + \frac{3 tan ^{10} ( θ )}{2} + \frac{tan ^{12} ( θ )}{2} + \frac{tan ^{14} ( θ )}{14}

定数を解答に追加する

= \frac{tan ^{2} ( θ )}{2} + \frac{3 tan ^{4} ( θ )}{2} + \frac{5 tan ^{6} ( θ )}{2} + \frac{5 tan ^{8} ( θ )}{2} + \frac{3 tan ^{10} ( θ )}{2} + \frac{tan ^{12} ( θ )}{2} + \frac{tan ^{14} ( θ )}{14} + C

グラフ

人気の例

integral of xcos(b)x

\int x cos (b) x d x

integral of 2/(sqrt(x)\sqrt{1-x)}

\int \frac{2}{x 1 - x} d x

integral of (cos^5(a))/(sqrt(sin(a)))

\int \frac{cos ^{5} ( a )}{sin ( a )} d a

integral of (x-1)/(4x^2+4x+5)

\int \frac{x - 1}{4 x ^{2} + 4 x + 5} d x

integral of 1/(sqrt(5x^2-x-1))

\int \frac{1}{5 x ^{2} - x - 1} d x