What is the integral of (e^{1/t})/(t^3) ?

The integral of (e^{1/t})/(t^3) is -(e^{1/t})/t+e^{1/t}+C

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integral of (e^{1/t})/(t^3)

\int \frac{e ^{\frac{1}{t}}}{t ^{3}} d t

- \frac{e ^{\frac{1}{t}}}{t} + e^{\frac{1}{t}} + C

Solution steps

\int \frac{e ^{\frac{1}{t}}}{t ^{3}} d t

Apply u-substitution

: \int - e^{u} u d u

= \int - e^{u} u d u

Take the constant out:

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

= - \int e^{u} u d u

Apply Integration By Parts

: e^{u} u - \int e^{u} d u

= - (e^{u} u - \int e^{u} d u)

\int e^{u} d u = e^{u}

= - (e^{u} u - e^{u})

Substitute back

u = \frac{1}{t}

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

Simplify

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

= - \frac{e ^{\frac{1}{t}}}{t} + e^{\frac{1}{t}}

Add a constant to the solution

= - \frac{e ^{\frac{1}{t}}}{t} + e^{\frac{1}{t}} + C

e^{y}

\int cos (x) (2 + sin (x))^{5} d x

x \to 0 lim (x^{2} + 2 x - 2)

\frac{\partial}{\partial x} (e^{x} ln (1 + y))

(x + 1) \frac{d y}{d x} + y = ln (x), y (1) = 10

What is the integral of (e^{1/t})/(t^3) ?
The integral of (e^{1/t})/(t^3) is -(e^{1/t})/t+e^{1/t}+C