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受欢迎的微积分 >

integral of (21x^2)/(sec(x^3-3))

解答

\int \frac{21 x ^{2}}{sec ( x ^{3} - 3 )} d x

解答

7 sin (x^{3} - 3) + C

求解步骤

\int \frac{21 x ^{2}}{sec ( x ^{3} - 3 )} d x

提出常数:

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

= 21 \cdot \int \frac{x ^{2}}{sec ( x ^{3} - 3 )} d x

使用基本三角恒等式:

\frac{1}{sec ( x )} = cos (x)

= 21 \cdot \int x^{2} cos (x^{3} - 3) d x

使用换元积分法

= 21 \cdot \int \frac{cos ( u )}{3} d u

提出常数:

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

= 21 \cdot \frac{1}{3} \cdot \int cos (u) d u

使用积分常见定则:

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

= 21 \cdot \frac{1}{3} sin (u)

u = x^{3} - 3

代回

= 21 \cdot \frac{1}{3} sin (x^{3} - 3)

化简

21 \cdot \frac{1}{3} sin (x^{3} - 3) : 7 sin (x^{3} - 3)

= 7 sin (x^{3} - 3)

解答补常数

= 7 sin (x^{3} - 3) + C

作图

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