defazajx

2021-10-15

Solve the integral.

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

diskusje5

Skilled2021-10-16Added 82 answers

Step 1

Given:$\int \frac{{v}^{2}dv}{{(49-{v}^{2})}^{\frac{3}{2}}}$

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,

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

substituting${(49-{v}^{2})}^{\frac{-1}{2}}=t$ and differentiating both sides w.r.t. v, we get,

$\frac{-1}{2}\cdot \frac{1}{{(49-{v}^{2})}^{\frac{3}{2}}}\cdot (-2v)dv=dt$

$\frac{1}{{(49-{v}^{2})}^{\frac{3}{2}}}dv=\frac{dt}{v}$

From the above expression, we can write as,

$=\int {v}^{2}\cdot \frac{dt}{{v}^{2}}$

$=\int dt$

=t+C

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

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

Answer-Hence the value of the integral$\int \frac{{v}^{2}dv}{{(49-{v}^{2})}^{\frac{3}{2}}}\text{}is\text{}\frac{1}{\sqrt{(49-{v}^{2})}}+C$ .

Given:

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,

substituting

From the above expression, we can write as,

=t+C

Answer-Hence the value of the integral

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