Question

Evaluate the integral. (use c for the constant of integration.) 3x 1 − x4 dx

Answer 1

The integral expression ∫3x(1 - x⁴), dx when evaluated is 3x²/2 - x⁶/2 + c

How to evaluate the integral expression

From the question, we have the following parameters that can be used in our computation:

∫3x(1 - x⁴) dx

Factor out 3 from the expression

So, we have

∫3x(1 - x⁴), dx = 3∫x(1 - x⁴), dx

Open the brackets

This gives

∫3x(1 - x⁴), dx = 3∫[x - x⁵], dx

Integrate the expression

So we have the following representation

∫3x(1 - x⁴), dx = 3[x²/2 - x⁶/6] + c

Open the brackets

This gives

∫3x(1 - x⁴), dx = 3x²/2 - x⁶/2 + c

Hence, the integral expression when evaluated is ∫3x(1 - x⁴), dx = 3x²/2 - x⁶/2 + c

Question

Evaluate the integral. (use c for the constant of integration).

∫3x(1 - x⁴) dx

Answer 2

To evaluate ∫3x(1 - x^4) dx, use the power rule for integration. The integral is
`(3/2)x^2 + x - (1/5)x^5 + C.`

Step-by-step explanation:

To evaluate the integral
`∫3x(1 - x^4) dx,`power rule for integration. The power rule states that the integral of
`x^n`
`(1/(n+1)) * x^(n+1) + C`t of integration.

Applying the power rule to each term of the integrand, we have:

The integral of 3x is
`(3/2)x^2 + C.`

The integral of
`(1 - x^4) is x - (1/5)x^5 + C.`

Combining these results, the integral of
`3x(1 - x^4) dx is (3/2)x^2 + x - (1/5)x^5 + C.`