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受欢迎的微积分 >

integral of 1/(sqrt(9+16x-4x^2))

解答

\int \frac{1}{9 + 16 x - 4 x ^{2}} d x

解答

\frac{1}{2} arcsin (\frac{2}{5} (x - 2)) + C

求解步骤

\int \frac{1}{9 + 16 x - 4 x ^{2}} d x

对

9 + 16 x - 4 x^{2}

配方:

- 4 (x - 2)^{2} + 25

= \int \frac{1}{- 4 ( x - 2 ) ^{2} + 25} d x

使用换元积分法

= \int \frac{1}{- 4 u ^{2} + 25} d u

使用三角换元法

= \int \frac{1}{2} d v

提出常数:

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

= \frac{1}{2} \cdot \int 1 d v

常数积分:

\int a d x = a x

= \frac{1}{2} \cdot 1 \cdot v

代回

= \frac{1}{2} \cdot 1 \cdot arcsin (\frac{2}{5} (x - 2))

化简

= \frac{1}{2} arcsin (\frac{2}{5} (x - 2))

解答补常数

= \frac{1}{2} arcsin (\frac{2}{5} (x - 2)) + C

作图

流行的例子

integral of (x^2)/(sqrt((x^2+1)^5))

\int \frac{x ^{2}}{( x ^{2} + 1 ) ^{5}} d x

integral of 45(1+x^2)(7+15x+5x^3)^{10}

\int 45 (1 + x^{2}) (7 + 15 x + 5 x^{3})^{10} d x

integral of (x^3-4x+6)/(x+7)

\int \frac{x ^{3} - 4 x + 6}{x + 7} d x

integral of (2sqrt(x))^2

\int (2 x)^{2} d x

integral of (x^2+57)/(x^2+64)

\int \frac{x ^{2} + 57}{x ^{2} + 64} d x