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受欢迎的微积分 >

积分 1/((x^2+0.75))

解答

积分

\frac{1}{( x ^{2} + 0.75 )}

解答

1.15470 \dots arctan (1.15470 \dots x) + C

求解步骤

\int \frac{1}{x ^{2} + 0.75} d x

x^{2} + 0.75 = 0.75 (\frac{x ^{2}}{0.75} + 1)

= \int \frac{1}{0.75 ( \frac{x ^{2}}{0.75} + 1 )} d x

提出常数:

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

= \frac{1}{0.75} \cdot \int \frac{1}{\frac{x ^{2}}{0.75} + 1} d x

化简

= 1.33333 \dots \cdot \int \frac{1}{\frac{x ^{2}}{0.75} + 1} d x

使用换元积分法

= 1.33333 \dots \cdot \int \frac{0.86602 \dots}{u ^{2} + 1} d u

提出常数:

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

= 1.33333 \dots \cdot 0.86602 \dots \cdot \int \frac{1}{u ^{2} + 1} d u

使用积分常见定则:

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

= 1.33333 \dots \cdot 0.86602 \dots arctan (u)

u = 1.15470 \dots x

代回

= 1.33333 \dots \cdot 0.86602 \dots arctan (1.15470 \dots x)

化简

= 1.15470 \dots arctan (1.15470 \dots x)

解答补常数

= 1.15470 \dots arctan (1.15470 \dots x) + C

作图

流行的例子

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