Get the answer to "Solve the integral. ∫v2dv(49−v2)32"

defazajx

Answered question

2021-10-15

Solve the integral.

\int \frac{v^{2} d v}{{(49 - v^{2})}^{\frac{3}{2}}}

Answer & Explanation

diskusje5

Skilled2021-10-16Added 82 answers

Step 1
Given:

\int \frac{v^{2} d v}{{(49 - v^{2})}^{\frac{3}{2}}}

To find- The value of the above integral.
Concept Used- Above integral can be evaluated by substitution method.
Step 2
Explanation- Rewrite the given expression as,

I = \int \frac{v^{2} d v}{{(49 - v^{2})}^{\frac{3}{2}}}

substituting

{(49 - v^{2})}^{\frac{- 1}{2}} = t

and differentiating both sides w.r.t. v, we get,

\frac{- 1}{2} \cdot \frac{1}{{(49 - v^{2})}^{\frac{3}{2}}} \cdot (- 2 v) d v = d t

\frac{1}{{(49 - v^{2})}^{\frac{3}{2}}} d v = \frac{d t}{v}

From the above expression, we can write as,

= \int v^{2} \cdot \frac{d t}{v^{2}}

= \int d t

=t+C

= {(49 - v^{2})}^{\frac{- 1}{2}} + C

= \frac{1}{\sqrt{(49 - v^{2})}} + C

Answer-Hence the value of the integral

\int \frac{v^{2} d v}{{(49 - v^{2})}^{\frac{3}{2}}} i s \frac{1}{\sqrt{(49 - v^{2})}} + C

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