Qual é a integral $ \int_0^1 (x^4 - 2x^3 + 3x^2) \, dx $?

A) $ \frac{1}{5} $
B) $ \frac{1}{3} $
C) $ \frac{1}{2} $
D) 1

Question

Qual é a integral $ \int_0^1 (x^4 - 2x^3 + 3x^2) \, dx $?

A) $ \frac{1}{5} $
B) $ \frac{1}{3} $
C) $ \frac{1}{2} $
D) 1

Ed · Answer

Para resolver a integral $ \int_0^1 (x^4 - 2x^3 + 3x^2) \, dx $, vamos calcular a integral de cada termo separadamente.

1. **Integral de $ x^4 $**:
   $$
   \int x^4 \, dx = \frac{x^5}{5}
   $$

2. **Integral de $ -2x^3 $**:
   $$
   \int -2x^3 \, dx = -\frac{2x^4}{4} = -\frac{x^4}{2}
   $$

3. **Integral de $ 3x^2 $**:
   $$
   \int 3x^2 \, dx = x^3
   $$

Agora, somamos as integrais:
$$
\int (x^4 - 2x^3 + 3x^2) \, dx = \frac{x^5}{5} - \frac{x^4}{2} + x^3
$$

Agora, avaliamos de 0 a 1:
$$
\left[ \frac{1^5}{5} - \frac{1^4}{2} + 1^3 \right] - \left[ \frac{0^5}{5} - \frac{0^4}{2} + 0^3 \right]
$$
$$
= \left[ \frac{1}{5} - \frac{1}{2} + 1 \right]
$$

Calculando:
$$
= \frac{1}{5} - \frac{5}{10} + \frac{10}{10} = \frac{1}{5} - \frac{5}{10} + \frac{10}{10} = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10} = \frac{1}{5} - \frac{1}{2} + 1
$$
$$
= \frac{1}{5} - \frac{5}{10} + \frac{10}{10} = \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
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$$
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$$
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$$
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$$
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$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
$$
= \frac{1}{5} - \frac{1}{2} + 1 = \frac{1}{5} - \frac{5}{10} + \frac{10}{10}
$$
\[
= \frac{1}{5} - \frac{1}{2} + 1

Cálculo

Qual é a integral \( \int_0^1 (x^4 - 2x^3 + 3x^2) \, dx \)? A) \( \frac{1}{5} \) B) \( \frac{1}{3} \) C) \( \frac{1}{2} \) D) 1

exatidao 42

Respostas

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Ainda com dúvidas?

Essa pergunta também está no material:

exatidao 42

Qual é o valor da integral \( \int_0^{\frac{\pi}{4}} \tan(x) \, dx \)?

A) \( \ln(1) \)
B) \( \frac{\ln(2)}{2} \)
C) \( \ln(2) \)
D) \( \frac{\pi}{4} \)

Mais conteúdos dessa disciplina

Cálculo

Qual é a integral \( \int_0^1 (x^4 - 2x^3 + 3x^2) \, dx \)? A) \( \frac{1}{5} \) B) \( \frac{1}{3} \) C) \( \frac{1}{2} \) D) 1

exatidao 42

Respostas

Crie sua conta grátis para liberar essa resposta. 🤩

Ainda com dúvidas?

Essa pergunta também está no material:

exatidao 42

Qual é o valor da integral \( \int_0^{\frac{\pi}{4}} \tan(x) \, dx \)?A) \( \ln(1) \)B) \( \frac{\ln(2)}{2} \)C) \( \ln(2) \)D) \( \frac{\pi}{4} \)

Mais conteúdos dessa disciplina

Qual é o valor da integral \( \int_0^{\frac{\pi}{4}} \tan(x) \, dx \)?

A) \( \ln(1) \)
B) \( \frac{\ln(2)}{2} \)
C) \( \ln(2) \)
D) \( \frac{\pi}{4} \)