Question - Solving Integral of a Polynomial Function
mathantiderivative of polynomialintegral of polynomial functionsolving integralsintegration with respect to x
Solution:
The image shows an integral that you'd like to solve:$$\int (8x^3 - x^2 + 5x - 1) \, dx$$To solve this integral, integrate each term individually with respect to $$ x $$. When integrating a polynomial term of the form $$ x^n $$, the antiderivative is $$ \frac{x^{n+1}}{n+1} $$ provided that $$ n \neq -1 $$.$$\int 8x^3 \, dx = \frac{8}{4}x^{4} = 2x^4$$$$\int (-x^2) \, dx = -\frac{1}{3}x^{3}$$$$\int 5x \, dx = \frac{5}{2}x^{2}$$$$\int (-1) \, dx = -x$$Now summing all of the individual antiderivatives and adding a constant of integration $$ C $$, we have:$$2x^4 - \frac{1}{3}x^{3} + \frac{5}{2}x^{2} - x + C$$So the integral of $$ 8x^3 - x^2 + 5x - 1 $$ with respect to $$ x $$ is:$$\int (8x^3 - x^2 + 5x - 1) \, dx = 2x^4 - \frac{1}{3}x^{3} + \frac{5}{2}x^{2} - x + C$$
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