Question

can someone do this problem:
integrate: ((5x+8)^1/2)+11)/sqrt5x+8, without substitution

Answer 1

Still tilting at windmills eh?

Answer 2

i know this problem can be solved witout substituiton

Answer 3

what is the final answer to this problem?

Answer 4

\[\int \frac{\sqrt{5x+8}+11}{\sqrt{5x+8}}dx\] ?

Answer 5

i swear i have done more complicated algebra today than in the last ten years at least.

Answer 6

what does it mean "without substitution"? are you supposed to look at this and say "the anti-derivaitve is.."?

Answer 7

\[\int {\sqrt{5x+8} + 11 \over \sqrt{5x+8}}dx\]\[= \int 1 + \frac{11}{\sqrt{5x+8}}dx\]\[=\int 1dx + \int \frac{11}{\sqrt{5x+8}}dx\]\[= x + 11\int\frac{1}{\sqrt{5x+8}}dx\]
Let \( u = 5x + 8 \implies dx = \frac{1}{5}du\)
\[\implies x + 11\int\frac{1}{\sqrt{5x+8}}dx\]\[= x+ 11\int \frac{1}{\sqrt{u}}(\frac{1}{5})du\]\[= x + \frac{11}{5}\int\frac{1}{\sqrt{u}}du\]\[= x+\frac{22}{5}\sqrt{5x+8} + C\]