Question - Integration of a Polynomial
Solution:
Certainly! You are asked to integrate the given polynomial. Here’s how you do it:The integral of a polynomial is found by integrating each term individually. The integral of $$ x^n $$ with respect to $$ x $$ is $$ \frac{x^{n+1}}{n+1} $$ provided $$ n \neq -1 $$.Let’s integrate each term of the given polynomial:$$\int (8x^3 - x^2 + 5x - 1) dx$$Integrating term by term:$$= 8 \int x^3 dx - \int x^2 dx + 5 \int x dx - \int 1 dx$$Applying the power rule for integration:$$= 8 \cdot \frac{x^{3+1}}{3+1} - \frac{x^{2+1}}{2+1} + 5 \cdot \frac{x^{1+1}}{1+1} - x + C$$Simplifying the expressions:$$= 8 \cdot \frac{x^4}{4} - \frac{x^3}{3} + 5 \cdot \frac{x^2}{2} - x + C$$Further simplifying, which also involves reducing the fraction $$8 \cdot \frac{x^4}{4}$$ to $$2x^4$$:$$= 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C$$So the integral of $$ 8x^3 - x^2 + 5x - 1 $$ with respect to $$ x $$ is $$ 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C $$, where $$ C $$ is the constant of integration.
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