Exercícios Resolvidos de Substituição Simples

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Exercício 158123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257258259260261262263264265266267268269270271272273274275276277278279280281282283284285286287288289290291292293294295296297298299300301302303304305306307308309310311312313314315316317318319320321322323324325326327328329330331332333334335336337338339340341342343344345346347348349350351352353354355356357358359360361362363364365366367368369370371372373374375376377378379380381382383384385386387388389390391392393394395396397398399400401402403404405406407408409410411412413414415416417418419420421422423424425426427428429430431432433434435436437438439440441442443444445446447448449450451452453454455456457458459460461462463464465466467468469470471472473474475476477478479480481482483484485486487488489490491492493494495496497498499500501502503504505506507508509510511512513514515516517518519520521522523524525526527528529530531532533534535536537538539540541542543544545546547548549550551552553554555556557558559560561562563564565566567568569570571572573574575576577578579580581582583584585586587588589590591592593594595596597598599600601602603604605

Enunciado

Calcule:

lim n → ∞ ⁡ ∑ i = 1 n 7 n ⋅ 3 i n + 1

Passo 1

Vamos tentar escrever esse limite na forma de uma integral na forma:

∫ a b f x d x

Precisamos achar a , b e f x .

Para tal, precisamos escrever o limite na forma: l i m n → ∞ ∑ i = 1 n f a + i Δ x Δ x

Vamos tentar pegar essa parte que está fora da raiz e usá-la como Δ x : Δ x = 7 n = b - a n

Passo 2

Considerando b = 8 e a = 1 , satisfazemos e equação acima e obtemos:

f a + i Δ x = f 1 + i Δ x = f 1 + 7 i n

Considerando o argumento da função como x : x = 1 + 7 i n ⟹ x = 7 i + n n ⟹ 7 i + n = x n ⟹ i = x n - n 7

Observando o limite:

l i m n → ∞ ∑ i = 1 n f a + i Δ x Δ x = lim n → ∞ ⁡ ∑ i = 1 n 7 n ⋅ 3 i n + 1 = lim n → ∞ ⁡ ∑ i = 1 n 3 i n + 1 ⋅ Δ x

Comparando, vemos que:

f 1 + 7 i n = 3 i n + 1

Substituindo o valor de i :

f x = 3 n ⋅ x - 1 n 7 + 1 = 3 7 x - 1 + 1 = 3 7 x - 3 7 + 1 = 3 7 x + 4 7

Passo 3

Pronto, então temos todos os elementos de uma integral. Podemos escrevê-la.

lim n → ∞ ⁡ ∑ i = 1 n 7 n ⋅ 3 i n + 1 = ∫ 1 8 3 7 x + 4 7 d x

Para resolver o limite, resolvemos a integral. Para tal, consideramos: u = 3 7 x + 4 7

Levando a:

d u = 3 7 d x u m i n = 1 ,       u m a x = 4

∫ 1 8 3 7 x + 4 7 d x = 7 3 ∫ 1 4 u d u = 7 3 ∫ 1 4 u 1 / 2 d u

= 7 3 ⋅ u 3 / 2 3 / 2 1 4 = 14 9 8 - 1 = 98 9

Resposta

lim n → ∞ ⁡ ∑ i = 1 n 7 n ⋅ 3 i n + 1 = 98 9

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