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受欢迎的微积分 >

integral of 1/(sqrt(16x^2+25))

解答

\int \frac{1}{16 x ^{2} + 25} d x

解答

\frac{1}{4} ln (\frac{4 x + 25 + 16 x ^{2}}{5}) + C

求解步骤

\int \frac{1}{16 x ^{2} + 25} d x

使用三角换元法

= \int \frac{sec ( u )}{4} d u

提出常数:

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

= \frac{1}{4} \cdot \int sec (u) d u

使用积分常见定则:

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= \frac{1}{4} ln ∣ tan (u) + sec (u) ∣

u = arctan (\frac{4}{5} x)

代回

= \frac{1}{4} ln tan (arctan (\frac{4}{5} x)) + sec (arctan (\frac{4}{5} x))

化简

\frac{1}{4} ln tan (arctan (\frac{4}{5} x)) + sec (arctan (\frac{4}{5} x)) : \frac{1}{4} ln (\frac{4 x + 25 + 16 x ^{2}}{5})

= \frac{1}{4} ln (\frac{4 x + 25 + 16 x ^{2}}{5})

解答补常数

= \frac{1}{4} ln (\frac{4 x + 25 + 16 x ^{2}}{5}) + C

作图

流行的例子

integral of sqrt(15-2x-x^2)

\int 15 - 2 x - x^{2} d x

(dy)/(dt)+k(340-y)=0

\frac{d y}{d t} + k (340 - y) = 0

laplacetransform e

l a pl a ce t r an s f or m e

integral of-4x^2(4x^3-1)^7

\int - 4 x^{2} (4 x^{3} - 1)^{7} d x

f (x) = x^{2 x}