integral of sin^{10}(θ)cos^3(θ)

解答

\int sin^{10} (θ) cos^{3} (θ) d θ

\frac{sin ^{11} ( θ )}{11} - \frac{sin ^{13} ( θ )}{13} + C

求解步骤

\int sin^{10} (θ) cos^{3} (θ) d θ

使用三角恒等式改写

= \int (1 - sin^{2} (θ)) cos (θ) sin^{10} (θ) d θ

使用换元积分法

= \int u^{10} (1 - u^{2}) d u

乘开

u^{10} (1 - u^{2}) : u^{10} - u^{12}

= \int u^{10} - u^{12} d u

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u^{10} d u - \int u^{12} d u

\int u^{10} d u = \frac{u ^{11}}{11}

\int u^{12} d u = \frac{u ^{13}}{13}

= \frac{u ^{11}}{11} - \frac{u ^{13}}{13}

u = sin (θ)

代回

= \frac{sin ^{11} ( θ )}{11} - \frac{sin ^{13} ( θ )}{13}

解答补常数

= \frac{sin ^{11} ( θ )}{11} - \frac{sin ^{13} ( θ )}{13} + C

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

\int \frac{1}{( 5 x - 2 ) ^{\frac{2}{3}}} d x

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

\int \frac{x + 7}{x ^{2} + 25} d x

x \to 0 lim (\frac{1 + x}{x})