I=∫V+V^2V+V^3 dV I=∫1+V1+V^2 dV I=∫11+V^2 dV+∫V1+V^2dV I=∫11+V^2 dV+12∫2V1+V^2dV I=tan^ 1 V+12ln (1+V^2)+C Hence, this is the answer. ...

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∫V + V2V + V3...

Question

$\int \frac{V + V^{2}}{V + V^{3}} d V =$ .

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Solution

$I = \int \frac{V + V^{2}}{V + V^{3}} d V$

$I = \int \frac{1 + V}{1 + V^{2}} d V$

$I = \int \frac{1}{1 + V^{2}} d V + \int \frac{V}{1 + V^{2}} d V$

$I = \int \frac{1}{1 + V^{2}} d V + \frac{1}{2} \int \frac{2 V}{1 + V^{2}} d V$

$I = {tan}^{- 1} V + \frac{1}{2} ln (1 + V^{2}) + C$

Hence, this is the answer.

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