Question

Enter the exact value with the simplest fraction form. Rounded decimal number is not accepted.

Answer 1

The value of
`\(\int_(0)^(1) (5x^3 + 2x^4) \,dx\) is \((33)/(20)\).`

To evaluate the given integral
`\(\int_(0)^(1) (5x^3 + 2x^4) \,dx\),`we can use the properties of definite integrals and the linearity of integration. We'll break the integral into two separate integrals:

`\[ \int_(0)^(1) (5x^3 + 2x^4) \,dx = \int_(0)^(1) 5x^3 \,dx + \int_(0)^(1) 2x^4 \,dx \]`

Now, we'll find the antiderivatives of \(5x^3\) and \(2x^4\) and evaluate them at the upper and lower limits:

`\[ \int_(0)^(1) 5x^3 \,dx = (5)/(4)x^4 \Big|_(0)^(1) = (5)/(4) \]`

`\[ \int_(0)^(1) 2x^4 \,dx = (2)/(5)x^5 \Big|_(0)^(1) = (2)/(5) \]`

Now, add these results together:

`\[ (5)/(4) + (2)/(5) = (25)/(20) + (8)/(20) = (33)/(20) \]`

Therefore, the value of
`\(\int_(0)^(1) (5x^3 + 2x^4) \,dx\) is \((33)/(20)\).`