Encontre a primitiva mais geral de cada uma das seguintes funções:
f(t) = 2 * raiz(t) + 6 * cos(t).

Passo 1

Beleza, vamos resolver esse exercício de cálculo! A gente precisa encontrar a primitiva mais geral da função . Isso quer dizer que vamos calcular a antiderivada dessa função, ou seja, encontrar uma função cuja derivada seja . Para isso, vamos precisar usar as regras de integração para funções como raiz quadrada e cosseno.

Passo 2

Primeiro, vamos separar a integral em duas partes, porque a antiderivada de uma soma é a soma das antiderivadas. Então temos:

Passo 3

Vamos resolver a primeira integral: . Lembre-se que . A regra para integrar é aumentar o expoente em 1 e dividir pelo novo expoente. Então:

Passo 4

Agora, vamos para a segunda integral: . A antiderivada de é , então:

Resposta

Juntando as duas partes, a primitiva mais geral de é:
onde é a constante de integração.

Exercícios relacionados

<html><head></head><body><div>Nos Problemas 1-12, ache a solução geral do sistema dado. <span class="math-inline "></span></div>
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<html><head></head><body><div>Nos Problemas 1-12, ache a solução geral do sistema dado. <span class=“math-inline “>
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Calcule as integrais:
(g) ;
Calcule as integrais:
(h)