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人気のある微分積分 >

integral of cos(x)sqrt(9+25sin^2(x))

解

\int cos (x) 9 + 25 sin^{2} (x) d x

解

\frac{1}{2} sin (x) 25 sin^{2} (x) + 9 + \frac{9}{10} ln (\frac{1}{3} 5 sin (x) + 9 + 25 sin^{2} (x)) + C

解答ステップ

\int cos (x) 9 + 25 sin^{2} (x) d x

u置換積分法を適用する

= \int 9 + 25 u^{2} d u

三角関数による置換を適用する

= \int \frac{9}{5} sec^{3} (v) d v

定数を除く:

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

= \frac{9}{5} \cdot \int sec^{3} (v) d v

積分換算を適用する

= \frac{9}{5} (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} \cdot \int sec (v) d v)

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= \frac{9}{5} (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

代用を戻す

= \frac{9}{5} (\frac{sec ^{2} ( arctan ( \frac{5}{3} sin ( x ) ) ) sin ( arctan ( \frac{5}{3} sin ( x ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{5}{3} sin (x))) + sec (arctan (\frac{5}{3} sin (x))))

簡素化

\frac{9}{5} (\frac{sec ^{2} ( arctan ( \frac{5}{3} sin ( x ) ) ) sin ( arctan ( \frac{5}{3} sin ( x ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{5}{3} sin (x))) + sec (arctan (\frac{5}{3} sin (x)))) : \frac{1}{2} sin (x) 25 sin^{2} (x) + 9 + \frac{9}{10} ln (\frac{1}{3} 5 sin (x) + 9 + 25 sin^{2} (x))

= \frac{1}{2} sin (x) 25 sin^{2} (x) + 9 + \frac{9}{10} ln (\frac{1}{3} 5 sin (x) + 9 + 25 sin^{2} (x))

定数を解答に追加する

= \frac{1}{2} sin (x) 25 sin^{2} (x) + 9 + \frac{9}{10} ln (\frac{1}{3} 5 sin (x) + 9 + 25 sin^{2} (x)) + C

グラフ

人気の例

integral of cot^2(ax)

\int cot^{2} (a x) d x

integral of (x-2)/(\sqrt[3]{x)}

\int \frac{x - 2}{3 x} d x

integral of (3x)/((x+2)(x+3))

\int \frac{3 x}{( x + 2 ) ( x + 3 )} d x

integral of cos(3z)sin^{10}(3z)

\int cos (3 z) sin^{10} (3 z) d z

integral of (sin(2x)+5tan^2(x)csc^2(x))

\int (sin (2 x) + 5 tan^{2} (x) csc^{2} (x)) d x