How do solve the following integral? int(t+7)^2/t^3dt

int(t+7)^2/t^3dt

2 Answers
Dec 4, 2017

Just expand the function and integrate. The answer is lnt-14/t-49/(2t^2)+C.

Explanation:

Let f(t)=(t+7)^2/t^3. First, let's rearrange f(t) into simpler terms.
f(t)=(t^2+14t+49)/t^3
=1/t+14/t^2+49/t^3
=t^-1+14t^-2+49t^-3

Then integrate f(t). Note that intx^ndx=1/(n+1)x^(n+1)+C if n!=-1, where C is the integral constant. If n=-1, intx^-1dx=int(1/x)dx=lnx +C is applied.

intf(t)dt=int(t^-1+14t^-2+49t^-3)dt
=intt^-1dt+14intt^-2dt+49intt^-3dt
=lnt+14(-t^-1)+49(-1/2t^-2)+C
=lnt-14/t-49/(2t^2)+C.

Dec 4, 2017

ln(t)-(7(4t+7))/(2t^2)+c

Explanation:

solving (t+7)^2

(t+7)^2=t^2+14t+49

now in the integral

int (t^2+14t+49)/t^3 dt

separating the fractions

int (t^2/t^3+14t/t^3+49/t^3) dt

int 1/t+14/t^2+49/t^3 dt

separting

int 1/tdt=ln(t)+c

int 14/t^2dt=-14/t+c

int49/t^3dt=-49/(2t^2)+c

int 1/t+14/t^2+49/t^3 dt=ln(t)-14/t-49/(2t^2)+c

operating

int 1/t+14/t^2+49/t^3 dt=ln(t)-(7(4t+7))/(2t^2)+c

int (t^2+14t+49)/t^3 dt=ln(t)-(7(4t+7))/(2t^2)+c