integral of (x^3-a^3)^2

解答

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

\frac{x ^{7}}{7} - \frac{a ^{3} x ^{4}}{2} + a^{6} x + C

求解步骤

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

乘开

(x^{3} - a^{3})^{2} : x^{6} - 2 a^{3} x^{3} + a^{6}

= \int x^{6} - 2 a^{3} x^{3} + a^{6} 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 x^{6} d x - \int 2 a^{3} x^{3} d x + \int a^{6} d x

\int x^{6} d x = \frac{x ^{7}}{7}

\int 2 a^{3} x^{3} d x = \frac{a ^{3} x ^{4}}{2}

\int a^{6} d x = a^{6} x

= \frac{x ^{7}}{7} - \frac{a ^{3} x ^{4}}{2} + a^{6} x

解答补常数

= \frac{x ^{7}}{7} - \frac{a ^{3} x ^{4}}{2} + a^{6} x + C

\int \frac{1}{1000 x - x ^{2}} d x

\int cos (2 x - 4) d x

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

\int (6 x^{2} - 5) e^{2 x^{3} - 5 x} d x

\int (5 x + 16)^{4} d x