ʃ (x4 + 6x3 – x2 + 2)dx
I really need it for practicing this subject...please please help
To solve the given integral, ʃ (x^4 + 6x^3 – x^2 + 2)dx, we need to apply the power rule of integration. The power rule states that if the term has the form x^n, where n is a constant (not equal to -1), then the integral is given by (1/(n+1)) * x^(n+1) + C, where C is the constant of integration.
Let's go ahead and apply this rule to each of the terms in the integrand:
ʃ x^4 dx = (1/5) * x^(4+1) + C = (1/5) * x^5 + C
ʃ 6x^3 dx = 6 * (1/4) * x^(3+1) + C = (3/2) * x^4 + C
ʃ -x^2 dx = -1 * (1/3) * x^(2+1) + C = (-1/3) * x^3 + C
ʃ 2 dx = 2x + C
Now we can substitute these results back into the original integral:
ʃ (x^4 + 6x^3 – x^2 + 2)dx
= ʃ x^4 dx + ʃ 6x^3 dx - ʃ x^2 dx + ʃ 2 dx
= (1/5) * x^5 + C + (3/2) * x^4 + C - (1/3) * x^3 + C + 2x + C
Combining like terms, we get:
(1/5) * x^5 + (3/2) * x^4 - (1/3) * x^3 + 2x + 4C
Therefore, the antiderivative of the given expression is (1/5) * x^5 + (3/2) * x^4 - (1/3) * x^3 + 2x + 4C.