积分 (2^x)/(2^x(1+4^x))

解答

积分

\frac{2 ^{x}}{2 ^{x} ( 1 + 4 ^{x} )}

- \frac{1}{2 ln ( 2 )} ln ∣ 1 + 4^{x} ∣ + x + C

求解步骤

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

约分:

2^{x}

= \int \frac{1}{1 + 4 ^{x}} d x

\frac{1}{1 + 4 ^{x}} = - \frac{4 ^{x}}{1 + 4 ^{x}} + 1

= \int - \frac{4 ^{x}}{1 + 4 ^{x}} + 1 d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int \frac{4 ^{x}}{1 + 4 ^{x}} d x + \int 1 d x

\int \frac{4 ^{x}}{1 + 4 ^{x}} d x = \frac{1}{2 ln ( 2 )} ln ∣ 1 + 4^{x} ∣

\int 1 d x = x

= - \frac{1}{2 ln ( 2 )} ln ∣ 1 + 4^{x} ∣ + x

解答补常数

= - \frac{1}{2 ln ( 2 )} ln ∣ 1 + 4^{x} ∣ + x + C

in t e g r a l \frac{e ^{- x}}{x ^{\frac{1}{2}}}

in t e g r a l cos^{2} (x) sin^{2} (x)

in t e g r a l (x - 2)^{5}

in t e g r a l x^{3} cos (x^{4})

in t e g r a l 1 + tan^{2} (x)