Question

integration of 1+v/2+2v-v² dv​

Answer 1

Answer:

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∫

v

2

+2v+1

v

2

dv

⇒∫

v

2

+2v+1

v

2

+(2v+1)−(2v+1)

dv

⇒∫

v

2

+2v+1

(v

2

+2v+1)−(2v+1)

dv=∫dv−∫

v

2

+2v+1

2v+1

dv

⇒v−∫

v

2

+2v+1

2v+1+(1−1)

dv

⇒v−∫

v

2

+2v+1

2v+2−1)

dv

⇒v−[∫

v

2

+2v+1

2v+2dv

−∫

v

2

+2v+1

dv

]

⇒v−[∫

v

2

+2v+1

(2v+2)dv

−∫

v

2

+2v+1

dv

]

Let v

2

+2v+1=t

(2v+2)dv =dt

⇒v−[∫

t

dt

−∫

(v+1)

2

dv

]

⇒v−ln∣t∣ +

v+1

−1

+c

⇒v−ln

∣

∣

∣

v

2

+2v+1

∣

∣

∣

-

v+1

1

+c