Question - Integration using Power Rule
Solution:
The image shows an integral expression that you'd like to evaluate. The integral is:∫ (8x^3 - x^2 + 5x - 1) dxTo 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 = -1xPutting it all together, the indefinite integral is:2x^4 - (1/3)x^3 + (5/2)x^2 - x + CWhere C is the constant of integration.
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