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受欢迎的微积分 >

integral from 0 to 1 of (x^2tan(x^3))/4

解答

\int_{0}^{1} \frac{x ^{2} tan ( x ^{3} )}{4} d x

解答

- \frac{1}{12} ln (cos (1))

+1

十进制

0.05130 \dots

求解步骤

\int_{0}^{1} \frac{x ^{2} tan ( x ^{3} )}{4} d x

提出常数:

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

= \frac{1}{4} \cdot \int_{0}^{1} x^{2} tan (x^{3}) d x

使用换元积分法

= \frac{1}{4} \cdot \int_{0}^{1} \frac{tan ( u )}{3} d u

提出常数:

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

= \frac{1}{4} \cdot \frac{1}{3} \cdot \int_{0}^{1} tan (u) d u

使用三角恒等式改写

= \frac{1}{4} \cdot \frac{1}{3} \cdot \int_{0}^{1} \frac{sin ( u )}{cos ( u )} d u

使用换元积分法

= \frac{1}{4} \cdot \frac{1}{3} \cdot \int_{1}^{c o s (1)} - \frac{1}{v} d v

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= \frac{1}{4} \cdot \frac{1}{3} (- \int_{c o s (1)}^{1} - \frac{1}{v} d v)

提出常数:

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

= \frac{1}{4} \cdot \frac{1}{3} (- (- \int_{c o s (1)}^{1} \frac{1}{v} d v))

使用积分常见定则:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{4} \cdot \frac{1}{3} (- (- [ln ∣ v ∣]_{c o s (1)}^{1}))

化简

\frac{1}{4} \cdot \frac{1}{3} (- (- [ln ∣ v ∣]_{c o s (1)}^{1})) : \frac{1}{12} [ln ∣ v ∣]_{c o s (1)}^{1}

= \frac{1}{12} [ln ∣ v ∣]_{c o s (1)}^{1}

带入上下限计算后得:

- ln (cos (1))

= \frac{1}{12} (- ln (cos (1)))

化简

= - \frac{1}{12} ln (cos (1))

作图

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