integral of cos(x^2)x^3

解答

\int cos (x^{2}) x^{3} d x

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

求解步骤

\int cos (x^{2}) x^{3} d x

使用换元积分法

= \int \frac{u cos ( u )}{2} d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int u cos (u) d u

使用分布积分法

= \frac{1}{2} (u sin (u) - \int sin (u) d u)

\int sin (u) d u = - cos (u)

= \frac{1}{2} (u sin (u) - (- cos (u)))

u = x^{2}

代回

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

化简

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

解答补常数

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

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

\int - 5 x d x

\int \frac{e ^{2 x}}{7 - e ^{4 x}} d x

\int \frac{4 x ^{3} - 8 x ^{2} + 4 x + 3}{4 x ^{2} - 8 x + 5} d x

\int (5 x^{4} - 3 x^{4} + 4 x^{2} - 2 + 6) d x