integral of xcot(x^2)sin(ln(sin(x^2)))

解答

\int x cot (x^{2}) sin (ln (sin (x^{2}))) d x

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

求解步骤

\int x cot (x^{2}) sin (ln (sin (x^{2}))) d x

使用换元积分法

= \int \frac{sin ( 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 sin (u) d u

使用积分常见定则:

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

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

u = ln (sin (x^{2}))

代回

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

化简

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

解答补常数

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

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