integral of 30sec^4(x)

解答

\int 30 sec^{4} (x) d x

30 (tan (x) + \frac{tan ^{3} ( x )}{3}) + C

求解步骤

\int 30 sec^{4} (x) d x

提出常数:

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

= 30 \cdot \int sec^{4} (x) d x

使用三角恒等式改写

= 30 \cdot \int (1 + tan^{2} (x)) sec^{2} (x) d x

使用换元积分法

= 30 \cdot \int 1 + u^{2} d u

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

= 30 (\int 1 d u + \int u^{2} d u)

\int 1 d u = u

\int u^{2} d u = \frac{u ^{3}}{3}

= 30 (u + \frac{u ^{3}}{3})

u = tan (x)

代回

= 30 (tan (x) + \frac{tan ^{3} ( x )}{3})

解答补常数

= 30 (tan (x) + \frac{tan ^{3} ( x )}{3}) + C

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