Evaluate-x-3 ex-x-53dx-

Let I=∫ (x+3) e^x/(x+5)^3dx =∫ (x+5 2) e^x/(x+5)^3dx =∫ e^x/(x+5)^2dx 2∫ e^x/(x+5)^3dx =∫ e^x (x+5)^ 2dx 2∫ e^x/(x+5)^3dx =(x+5)^ 2 e^x+2∫ (x+5)^ 3 e^x dx ...

You visited us 2 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Special Integrals - 2

Evaluate∫x+3 ...

Question

Evaluate

$\int \frac{(x + 3) e^{x}}{(x + 5)^{3}} d x$ .

Open in App

Solution

Let $I = \int \frac{(x + 3) e^{x}}{(x + 5)^{3}} d x$

$= \int \frac{(x + 5 - 2) e^{x}}{(x + 5)^{3}} d x$

$= \int \frac{e^{x}}{(x + 5)^{2}} d x - 2 \int \frac{e^{x}}{(x + 5)^{3}} d x$

$= \int e^{x} (x + 5)^{- 2} d x - 2 \int \frac{e^{x}}{(x + 5)^{3}} d x$

$= (x + 5)^{- 2} e^{x} + 2 \int (x + 5)^{- 3} e^{x} d x - 2 \int \frac{e^{x}}{(x + 5)^{3}} d x + c$

$= e^{x} (x + 5)^{- 2} + c = \frac{e^{x}}{(x + 5)^{2}} + c$

Suggest Corrections

Similar questions

Q. Evaluate

\int \frac{(x + 3) e^{x}}{(x + 5)^{3}} d x

Q. Evaluate:

\int \frac{(x + 3) e^{x}}{(x + 5)^{3}} d x

Q. Evaluate:

\int {(x^{2} + 5)}^{3} d x

Evaluate:

e^{(x - 1) - \frac{{(x - 1)}^{2}}{2} + \frac{{(x - 1)}^{3}}{3} - \frac{{(x - 1)}^{4}}{4} + . . . \infty}

Q. Evaluate

\int \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}} d x

Join BYJU'S Learning Program

Evaluate ∫(x+3) ex(x+5)3dx.

Let I=∫ (x+3) ex(x+5)3dx =∫ (x+5−2) ex(x+5)3dx =∫ ex(x+5)2dx−2∫ ex(x+5)3dx =∫ ex (x+5)−2dx−2∫ ex(x+5)3dx =(x+5)−2 ex+2∫ (x+5)−3 ex dx−2∫ ex(x+5)3dx+c =ex (x+5)−2+c=ex(x+5)2+c

Evaluate

$\int \frac{(x + 3) e^{x}}{(x + 5)^{3}} d x$ .