Expert answers to "What is the value of integral ∫dxx2+9"

Cameron Russell

Answered question

2022-01-25

What is the value of integral

\int \frac{d x}{\sqrt{x^{2} + 9}}

lilwhitelieyc

Beginner2022-01-26Added 10 answers

using the substitution

u = \frac{x}{3}

you get

\int \frac{d x}{\sqrt{x^{2} + 9}} = \frac{13}{\int} \frac{d x}{\sqrt{{(\frac{x}{3})}^{2} + 1}} = \int \frac{d u}{\sqrt{u^{2} + 1}} = \arcsin h (u) = \arcsin h (\frac{x}{3})

porekalahr

Beginner2022-01-27Added 10 answers

Do the substitution

x = 3 sinh t

and

d x = 3 cosh t d t

. Then your primitive becomes

\int \frac{3 cosh t}{\sqrt{9 {sinh}^{2} (t) + 9}} d t

Since

{sinh}^{2} (t) + 1 = {cosh}^{2} (t)

, (1) becomes

\int 1 d t = t = \arcsin h (\frac{x}{3})

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