(4x3-6x2+2x+5)dx

Suzhoumeite

3 min read

·

Jun 03, 2024

16

Share

** Integral of (4x^3 - 6x^2 + 2x + 5) dx **

In this article, we will discuss the integral of the function (4x^3 - 6x^2 + 2x + 5) with respect to x.

The Given Function

The given function is:

f(x) = 4x^3 - 6x^2 + 2x + 5

Our task is to find the integral of this function with respect to x, which is denoted by:

∫(4x^3 - 6x^2 + 2x + 5) dx

Solution

To find the integral, we will integrate each term separately and then combine the results.

Term 1: 4x^3

The integral of x^n is (x^(n+1))/(n+1). Therefore, the integral of 4x^3 is:

∫4x^3 dx = x^4 + C

Term 2: -6x^2

The integral of x^n is (x^(n+1))/(n+1). Therefore, the integral of -6x^2 is:

∫-6x^2 dx = -2x^3 + C

Term 3: 2x

The integral of x is (x^2)/2. Therefore, the integral of 2x is:

∫2x dx = x^2 + C

Term 4: 5

The integral of a constant is the constant multiplied by x. Therefore, the integral of 5 is:

∫5 dx = 5x + C

Combining the Results

Now, we combine the results of each term to obtain the final answer:

∫(4x^3 - 6x^2 + 2x + 5) dx = x^4 - 2x^3 + x^2 + 5x + C

where C is the constant of integration.

Therefore, the integral of (4x^3 - 6x^2 + 2x + 5) with respect to x is x^4 - 2x^3 + x^2 + 5x + C.