integral of sin^3(x)+cos^3(x)

解答

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

- cos (x) + \frac{cos ^{3} ( x )}{3} + sin (x) - \frac{sin ^{3} ( x )}{3} + C

求解步骤

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

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

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

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

= - cos (x) + \frac{cos ^{3} ( x )}{3} + sin (x) - \frac{sin ^{3} ( x )}{3}

解答补常数

= - cos (x) + \frac{cos ^{3} ( x )}{3} + sin (x) - \frac{sin ^{3} ( x )}{3} + C

\int \frac{4 x + 3}{x ^{2} + 1} d x

\int \frac{3}{( 3 x - 4 ) ln ( 3 x - 4 )} d x

\int e^{t - 1} d t

\int (x - 8) (x^{2} - 20 x + 96) d x

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