Question - Integration of Polynomials
mathpolynomial integration ruleintegration of polynomialsintegrating polynomialsintegral of polynomial functions
Solution:
To solve the given integral, integrate each term with respect to x:$$\int (8x^3 - x^2 + 5x - 1) \, dx$$Integrate term by term:$$= \int 8x^3 \, dx - \int x^2 \, dx + \int 5x \, dx - \int 1 \, dx$$Now apply the power rule of integration which is:$$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad \text{where} \quad n \neq -1$$So we get:$$= \frac{8x^{3+1}}{3+1} - \frac{x^{2+1}}{2+1} + \frac{5x^{1+1}}{1+1} - x + C$$Simplify the expression:$$= \frac{8x^4}{4} - \frac{x^3}{3} + \frac{5x^2}{2} - x + C$$$$= 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C$$Here, $$ C $$ is the constant of integration.So the final answer is:$$2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C$$
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com