Question

find the integral of the function with respect to x=e^5x+10​

Answer 1

Answer:

$\displaystyle \longrightarrow \frac{e^{5x}}{5}+10x +C$

Explanation:

Let :

$\displaystyle \text{I} = \int\limits{\left(e^{5x}+10\right)} \, dx$

$\displaystyle \text{I} = \int\limits{e^{5x}} \, dx+\int\limits{10} \, dx$

$\displaystyle \text{I} = \int\limits{e^{5x}} \, dx+10\int\limits{1} \, dx$

Substituting u = 5 x

Diff. w.r.t  x :

$\displaystyle \frac{du}{dx} =\frac{d}{dx} (5x)$

$\displaystyle \longrightarrow \frac{du}{dx} =5$

$\displaystyle \longrightarrow dx=\frac{du}{5}$

$\displaystyle \text{I} = \int\limits{e^{5x}} \, dx$

$\displaystyle \longrightarrow \text{I} = \frac{1}{5} \int\limits{e^{u}} \, du$

$\displaystyle \longrightarrow \text{I} = \frac{e^{u}}{5}$

$\displaystyle \longrightarrow \text{I} = \frac{e^{5x}}{5}$

$\displaystyle \text{I} =10\int\limits{1} \, dx$

$\displaystyle \longrightarrow \text{I} =10x$

$\displaystyle \longrightarrow \text{I} = \frac{e^{5x}}{5}+10x +C$

Hence we get required answer!