Integration using Power Rule
The image shows an integral expression that you'd like to evaluate. The integral is:
∫ (8x^3 - x^2 + 5x - 1) dx
To solve the integral, we use the power rule of integration which states that ∫ x^n dx = (x^(n+1))/(n+1) + C, where n is a real number different from -1, and C is the constant of integration.
Let's integrate each term individually:
∫ 8x^3 dx = (8x^(3+1))/(3+1) = (8x^4)/4 = 2x^4
∫ -x^2 dx = -(x^(2+1))/(2+1) = -(x^3)/3
∫ 5x dx = (5x^(1+1))/(1+1) = (5x^2)/2
∫ -1 dx = -1x
Putting it all together, the indefinite integral is:
2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
Where C is the constant of integration.
