What is $\int \frac{5}{{(1 + 3 x)}^{3}} d x$ $int (5)/(1+3x)^3dx$ ?

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Answer 1 · 2018-03-15T05:57:23

$- \frac{5}{6 {(1 + 3 x)}^{2}} + C$ $-5/{6(1+3x)^2}+C$

The integral $I = \int \frac{5}{{(1 + 3 x)}^{3}} d x$ $I=int5/(1+3x)^3dx$ , can be directly dealt with, if we

use the following Rule :

$intf(x)dx=F(x)+crArrintf(ax+b)dx=1/aF(ax+b)+c', (ane0).$

$f(x)=1/x^3=x^-3," we have, "F(x)=x^(-3+1)/(-3+1)=-1/(2x^2)$ .

$:. int5/(1+3x)^3dx=5int1/(1+3x)^3dx$ ,

$=5[1/3{-1/2*1/(1+3x)^2}]$ .

$rArr I=-5/6*1/(1+3x)^2+C$ .

OTHERWISE,

Let $1+3x=t," so that, "3dx=dt, or, dx=1/3dt$ .

$:. I=int5/(1+3x)^3dx=5int1/(1+3x)^3dx$ ,

$=5int1/t^3*1/3dt$ ,

$=5/3intt^-3dt$ ,

$=5/3*t^(-3+1)/(-3+1)$ ,

$=-5/6*t^-2,$

$=-5/(6t^2)$ .

$rArr I=-5/{6(1+3x)^2}+C$ , as before!

Enjoy Maths.!

What is int (5)/(1+3x)^3dx∫5(1+3x)3dxint (5)/(1+3x)^3dx?