Question text Find [math: ∫6x5+2x2−4x+5]\displaystyle\int{6\,x^5+2\,x^2-4\,x+5} [math: dx]. Note: Type "[math: c]" for the integral constant. [input] Check Question 15

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BlackTom AI · Accepted Answer

The task is to compute the indefinite integral of the polynomial 6x^5 + 2x^2 - 4x + 5 with respect to x.
First, integrate each term separately: for ∫6x^5 dx, the power rule gives x^6 since ∫x^n dx = x^{n+1}/(n+1); multiplying by 6 yields x^6.
Next, ∫2x^2 dx equals 2 * (x^3/3) = (2/3)x^3 by applying the same rule to x^2.
Then, ∫(-4x) dx is -4 * (x^2/2) = -2x^2, again using the power rule.
Finally, ∫5 dx is 5x, since the integral of a constant is that constant times x.
Combine all terms and add the constant of integration: x^6 + (2/3)x^3 - 2x^2 + 5x + C.
The provided answer, written as [math: x^6 + (2/3)x^3 - 2x^2 + 5x + c], matches this result, differing only in the conventional notation for the constant of integration (C vs c).
This aligns with standard antiderivative rules and the linearity of the integral, since each term was integrated independently and then summed.

Question text Find [math: ∫6x5+2x2−4x+5]\displaystyle\int{6\,x^5+2\,x^2-4\,x+5} [math: dx]. Note: Type "[math: c]" for the integral constant. [input] Check Question 15

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