defazajx

2021-10-15

Solve the integral.
$\int \frac{{v}^{2}dv}{{\left(49-{v}^{2}\right)}^{\frac{3}{2}}}$

diskusje5

Step 1
Given: $\int \frac{{v}^{2}dv}{{\left(49-{v}^{2}\right)}^{\frac{3}{2}}}$
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,
$I=\int \frac{{v}^{2}dv}{{\left(49-{v}^{2}\right)}^{\frac{3}{2}}}$
substituting ${\left(49-{v}^{2}\right)}^{\frac{-1}{2}}=t$ and differentiating both sides w.r.t. v, we get,
$\frac{-1}{2}\cdot \frac{1}{{\left(49-{v}^{2}\right)}^{\frac{3}{2}}}\cdot \left(-2v\right)dv=dt$
$\frac{1}{{\left(49-{v}^{2}\right)}^{\frac{3}{2}}}dv=\frac{dt}{v}$
From the above expression, we can write as,
$=\int {v}^{2}\cdot \frac{dt}{{v}^{2}}$
$=\int dt$
=t+C
$={\left(49-{v}^{2}\right)}^{\frac{-1}{2}}+C$
$=\frac{1}{\sqrt{\left(49-{v}^{2}\right)}}+C$
Answer-Hence the value of the integral .

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