Integration of x/(2x+5)?

2 Answers
Apr 7, 2018

I=1/4((2x+5)-5ln(abs((2x+5))))+C

Explanation:

We want to solve

I=intx/(2x+5)dx

Make a substitution color(blue)(u=2x+5=>du=2dx

I=1/2intx/udu

But color(blue)(u=2x+5=>x=(u-5)/2

I=1/2int((u-5)/2)/udu

color(white)(I)=1/4int(u-5)/udu

color(white)(I)=1/4int1-5/udu

color(white)(I)=1/4(u-5ln(abs(u)))+C

Substitute back u=(2x+5)

I=1/4((2x+5)-5ln(abs((2x+5))))+C

Apr 7, 2018

I=intx/(2x+5)dx

Let 2x+5 = t

Therefore, dt = 2 dx => dx=dt/2

Also, x=(t-5)/2

=>I=int(t-5)/(2(t)) dt/2

=>1/4int(t-5)/t dt

=>1/4[int1dt-5int1/t dt]

=>1/4[t-5log|t|] +c

=>1/4(2x+5)-5/4log|2x+5 | +c