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人気のある微分積分 >

integral of (sin(x))/((1-sin(x))^2)

解

\int \frac{sin ( x )}{( 1 - sin ( x ) ) ^{2}} d x

解

\frac{2}{3} sec^{3} (x) + \frac{2}{3} tan^{3} (x) - sec (x) + C

解答ステップ

\int \frac{sin ( x )}{( 1 - sin ( x ) ) ^{2}} d x

1 - sin (x)

の共役で乗じる

\frac{cos ^{2} ( x )}{1 + sin ( x )}

= \int \frac{sin ( x )}{( \frac{c o s ^{2} ( x )}{1 + s i n ( x )} ) ^{2}} d x

拡張

\frac{sin ( x )}{( \frac{c o s ^{2} ( x )}{1 + s i n ( x )} ) ^{2}} : \frac{sin ( x )}{cos ^{4} ( x )} + \frac{2 sin ^{2} ( x )}{cos ^{4} ( x )} + \frac{sin ^{3} ( x )}{cos ^{4} ( x )}

= \int \frac{sin ( x )}{cos ^{4} ( x )} + \frac{2 sin ^{2} ( x )}{cos ^{4} ( x )} + \frac{sin ^{3} ( x )}{cos ^{4} ( x )} d x

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 \frac{sin ( x )}{cos ^{4} ( x )} d x + \int \frac{2 sin ^{2} ( x )}{cos ^{4} ( x )} d x + \int \frac{sin ^{3} ( x )}{cos ^{4} ( x )} d x

\int \frac{sin ( x )}{cos ^{4} ( x )} d x = \frac{1}{3} sec^{3} (x)

\int \frac{2 sin ^{2} ( x )}{cos ^{4} ( x )} d x = \frac{2}{3} tan^{3} (x)

\int \frac{sin ^{3} ( x )}{cos ^{4} ( x )} d x = \frac{1}{3} sec^{3} (x) - sec (x)

= \frac{1}{3} sec^{3} (x) + \frac{2}{3} tan^{3} (x) + \frac{1}{3} sec^{3} (x) - sec (x)

類似した元を足す:

\frac{1}{3} sec^{3} (x) + \frac{1}{3} sec^{3} (x) = \frac{2}{3} sec^{3} (x)

= \frac{2}{3} sec^{3} (x) + \frac{2}{3} tan^{3} (x) - sec (x)

定数を解答に追加する

= \frac{2}{3} sec^{3} (x) + \frac{2}{3} tan^{3} (x) - sec (x) + C

グラフ

人気の例

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

\int \frac{x + 3}{x ^{2} + x + 1} d x

integral of (2x^2+4x-3)/((x-2)^3)

\int \frac{2 x ^{2} + 4 x - 3}{( x - 2 ) ^{3}} d x

integral of (x^2+6x-1)/(x^3-x)

\int \frac{x ^{2} + 6 x - 1}{x ^{3} - x} d x

integral of 1/(sqrt(1-y))

\int \frac{1}{1 - y} d y

integral of (-8x)/((x+1)-sqrt(x+1))

\int \frac{- 8 x}{( x + 1 ) - x + 1} d x