Exercícios Resolvidos de Substituição Simples

Ver Teoria
Exercício 861234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374757677787980818283848587888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257258259260261262263264265266267268269270271272273274275276277278279280281282283284285286287288289290291292293294295296297298299300301302303304305306307308309310311312313314315316317318319320321322323324325326327328329330331332333334335336337338339340341342343344345346347348349350351352353354355356357358359360361362363364365366367368369370371372373374375376377378379380381382383384385386387388389390391392393394395396397398399400401402403404405406407408409410411412413414415416417418419420421422423424425426427428429430431432433434435436437438439440441442443444445446447448449450451452453454455456457458459460461462463464465466467468469470471472473474475476477478479480481482483484485486487488489490491492493494495496497498499500501502503504505506507508509510511512513514515516517518519520521522523524525526527528529530531532533534535536537538539540541542543544545546547548549550551552553554555556557558559560561562563564565566567568569570571572573574575576577578579580581582583584585586587588589590591592593594595596597598599600601602603604605

Enunciado

Calcule as integrais dos exercícios 1 a 24 .

1 8 .   ∫ tan ⁡ x d x

Passo 1

Primeiramente, vamos reescrever esse termo assim:

∫ tan ⁡ x d x = ∫ s e n x cos ⁡ x d x

“Ah, mas por quê?” A gente já viu que:

d d x cos ⁡ x = - s e n ( x )

Será que aqui a gente consegue relacionar um com o outro nesta integral? Vamos tentar!

Passo 2

Chamando u = cos ⁡ x → d u = - s e n ( x ) d x . Substituindo na integral:

  ∫ s e n x cos ⁡ x d x → ∫ - 1 u d u

Olha só! Ficou mais fácil agora, né?

∫ - 1 u d u = - ∫ 1 u d u = - ln ⁡ u + C

Passo 3

Lembrando que a nossa variável NÃO é u , portanto, precisamos voltar pro que tínhamos antes:

u = c o s ⁡ ( x )

- ln ⁡ u + C = - ln ⁡ cos ⁡ x + C = ln ⁡ cos ⁡ x - 1 + C = ln ⁡ s e c ⁡ ( x ) + C

Resposta

1 8 .   ln ⁡ s e c ⁡ ( x ) + C

Exercícios de Livros Relacionados

Calcule a integral. ∫ x 3 x - 1 - 4 d x
Calcule a integral fazendo a substituição dada. ∫ cos ⁡ 3 x d x , u = 3 x
32. ∫ d t 2 + t 2
Use a fórmula de substituição do Teorema 7 para calcular as integrais nos Exercícios 1-46. ∫ π 3 π / 2 cot 5 ⁡ θ 6 sec 2 ⁡ θ 6 d θ
Calcule a integral ∫ sec ⁡ x cos ⁡ 2 x s e nx + sec ⁡ x d x