Finding an Indefinite Integral In Exercises 15–46, find the indefinite integral.

∫ [ ν + 1 ( 3 ν − 1 ) 3 ] d v

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To determine To calculate: The value of integral of the integral ∫ [ v + 1 ( 3 v − 1 ) 3 ] d v . Explanation Given: The provided expression is ∫ [ v + 1 ( 3 v − 1 ) 3 ] d v . Formula used: The integration rule is ∫ u n d u = u n + 1 n + 1 + C . Calculation: Consider the function ∫ [ v + 1 ( 3 v − 1 ) 3 ] d v . Simplify it, I = ∫ [ v + 1 ( 3 v − 1 ) 3 ] d v Write it as two integrals. I = ∫ v d v + ∫ 1 ( 3 v − 1 ) 3 d v Take ( 3 v − 1 ) in the numerator, I = ∫ v d v + ∫ ( 3 v − 1 ) − 3 d v …… (1) Let t = 3 v − 1 . So, differentiate above expression to get; d t = 3 d v So, the equation (1) gives; I = ∫ v d v + 1 3 ∫ ( t ) − 3 d t Recall the integration rule ∫ u n d u = u n + 1 n + 1 + C

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Finding an Indefinite Integral In Exercises 15–46, find the indefinite integral. ∫ [ ν + 1 ( 3 ν − 1 ) 3 ] d v

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Calculus Calculus 10th Edition

Finding an Indefinite Integral In Exercises 15–46, find the indefinite integral.

∫ [ ν + 1 ( 3 ν − 1 ) 3 ] d v