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人気のある微分積分 >

integral of (cos(θ))/(1+cos(θ))

解

\int \frac{cos ( θ )}{1 + cos ( θ )} d θ

解

- csc (θ) + cot (θ) + θ + C

解答ステップ

\int \frac{cos ( θ )}{1 + cos ( θ )} d θ

1 + cos (θ)

の共役で乗じる

\frac{sin ^{2} ( θ )}{1 - cos ( θ )}

= \int \frac{cos ( θ )}{\frac{s i n ^{2} ( θ )}{1 - c o s ( θ )}} d θ

拡張

\frac{cos ( θ )}{\frac{s i n ^{2} ( θ )}{1 - c o s ( θ )}} : \frac{cos ( θ )}{sin ^{2} ( θ )} - \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )}

= \int \frac{cos ( θ )}{sin ^{2} ( θ )} - \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )} d θ

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{cos ( θ )}{sin ^{2} ( θ )} d θ - \int \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )} d θ

\int \frac{cos ( θ )}{sin ^{2} ( θ )} d θ = - csc (θ)

\int \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )} d θ = - cot (θ) - θ

= - csc (θ) - (- cot (θ) - θ)

簡素化

= - csc (θ) + cot (θ) + θ

定数を解答に追加する

= - csc (θ) + cot (θ) + θ + C

グラフ

人気の例

integral of (x^2+x+6)/(x^3-8)

\int \frac{x ^{2} + x + 6}{x ^{3} - 8} d x

integral of sin(10x)sin(20x)

\int sin (10 x) sin (20 x) d x

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

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

integral of (sinh(4x))/(cosh(4x))

\int \frac{sinh ( 4 x )}{cosh ( 4 x )} d x

integral of ((-3)/x)

\int (\frac{- 3}{x}) d x