integral of (1/(e^{t/4)}-e^{t/4})^2

解答

\int (\frac{1}{e ^{\frac{t}{4}}} - e^{\frac{t}{4}})^{2} d t

\frac{t}{e ^{\frac{t}{2}}} - e^{- \frac{t}{2}} (t + 2) - 2 t + 2 e^{\frac{t}{2}} + C

求解步骤

\int (\frac{1}{e ^{\frac{t}{4}}} - e^{\frac{t}{4}})^{2} d t

化简

(\frac{1}{e ^{\frac{t}{4}}} - e^{\frac{t}{4}})^{2} : \frac{( 1 - e ^{\frac{t}{2}} ) ^{2}}{e ^{\frac{t}{2}}}

= \int \frac{( 1 - e ^{\frac{t}{2}} ) ^{2}}{e ^{\frac{t}{2}}} d t

乘开

\frac{( 1 - e ^{\frac{t}{2}} ) ^{2}}{e ^{\frac{t}{2}}} : \frac{1}{e ^{\frac{t}{2}}} - 2 + e^{\frac{t}{2}}

= \int \frac{1}{e ^{\frac{t}{2}}} - 2 + e^{\frac{t}{2}} d t

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{1}{e ^{\frac{t}{2}}} d t - \int 2 d t + \int e^{\frac{t}{2}} d t

\int \frac{1}{e ^{\frac{t}{2}}} d t = \frac{t}{e ^{\frac{t}{2}}} - e^{- \frac{t}{2}} (t + 2)

\int 2 d t = 2 t

\int e^{\frac{t}{2}} d t = 2 e^{\frac{t}{2}}

= \frac{t}{e ^{\frac{t}{2}}} - e^{- \frac{t}{2}} (t + 2) - 2 t + 2 e^{\frac{t}{2}}

解答补常数

= \frac{t}{e ^{\frac{t}{2}}} - e^{- \frac{t}{2}} (t + 2) - 2 t + 2 e^{\frac{t}{2}} + C

\int 10 - \frac{100}{x + 10} d x

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

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