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受欢迎的微积分 >

integral from-2 to 0 of sqrt(2x^2+3x+3)

解答

\int_{- 2}^{0} 2 x^{2} + 3 x + 3 d x

解答

\frac{3 6 + 5 10}{8 2} + \frac{15}{16 2} ln (\frac{15 + 6 10 + 10 6 + 8 15}{15})

+1

十进制

3.23079 \dots

求解步骤

\int_{- 2}^{0} 2 x^{2} + 3 x + 3 d x

对

2 x^{2} + 3 x + 3

配方:

2 (x + \frac{3}{4})^{2} + \frac{15}{8}

= \int_{- 2}^{0} 2 (x + \frac{3}{4})^{2} + \frac{15}{8} d x

使用换元积分法

= \int_{- \frac{5}{4}}^{\frac{3}{4}} \frac{16 u ^{2} + 15}{2 2} d u

提出常数:

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

= \frac{1}{2 2} \cdot \int_{- \frac{5}{4}}^{\frac{3}{4}} 16 u^{2} + 15 d u

使用三角换元法

= \frac{1}{2 2} \cdot \int_{- a r c t a n (\frac{5}{3})}^{a r c t a n (\frac{3}{5})} \frac{15}{4} sec^{3} (v) d v

提出常数:

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

= \frac{1}{2 2} \cdot \frac{15}{4} \cdot \int_{- a r c t a n (\frac{5}{3})}^{a r c t a n (\frac{3}{5})} sec^{3} (v) d v

使用降次积分法

= \frac{1}{2 2} \cdot \frac{15}{4} [\frac{sec ^{2} ( v ) sin ( v )}{2}]_{- a r c t a n (\frac{5}{3})}^{a r c t a n (\frac{3}{5})} + \frac{1}{2} \cdot \int_{- a r c t a n (\frac{5}{3})}^{a r c t a n (\frac{3}{5})} sec (v) d v

\int_{- a r c t a n (\frac{5}{3})}^{a r c t a n (\frac{3}{5})} sec (v) d v = ln (\frac{15 + 6 10 + 10 6 + 8 15}{15})

= \frac{1}{2 2} \cdot \frac{15}{4} [\frac{sec ^{2} ( v ) sin ( v )}{2}]_{- a r c t a n (\frac{5}{3})}^{a r c t a n (\frac{3}{5})} + \frac{1}{2} ln (\frac{15 + 6 10 + 10 6 + 8 15}{15})

化简

\frac{1}{2 2} \cdot \frac{15}{4} [\frac{sec ^{2} ( v ) sin ( v )}{2}]_{- a r c t a n (\frac{5}{3})}^{a r c t a n (\frac{3}{5})} + \frac{1}{2} ln (\frac{15 + 6 10 + 10 6 + 8 15}{15}) : \frac{15}{8 2} [\frac{1}{2} sec (v) tan (v)]_{- a r c t a n (\frac{5}{3})}^{a r c t a n (\frac{3}{5})} + \frac{1}{2} ln (\frac{15 + 6 10 + 10 6 + 8 15}{15})

= \frac{15}{8 2} [\frac{1}{2} sec (v) tan (v)]_{- a r c t a n (\frac{5}{3})}^{a r c t a n (\frac{3}{5})} + \frac{1}{2} ln (\frac{15 + 6 10 + 10 6 + 8 15}{15})

带入上下限计算后得:

\frac{6}{5} + \frac{10}{3}

= \frac{15}{8 2} (\frac{6}{5} + \frac{10}{3} + \frac{1}{2} ln (\frac{15 + 6 10 + 10 6 + 8 15}{15}))

化简

= \frac{3 6 + 5 10}{8 2} + \frac{15}{16 2} ln (\frac{15 + 6 10 + 10 6 + 8 15}{15})

作图

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