Question

Find an antiderivative of the following functions:

i) If f(x)=x^{7} then F(x)=
ii) If f(x)=x^{-3} then F(x)=
iii) If f(x)=3 x^{2}+3 x then F(x)=

Answer 1

The antiderivative of x^7 is (1/8)x^8 + C. The antiderivative of x^(-3) is -(1/2)x^(-2) + C. The antiderivative of 3x^2 + 3x is x^3 + (3/2)x^2 + C.

Step-by-step explanation:

i) To find the antiderivative of f(x) = x^7, we can use the power rule of integration. According to the power rule, if the function is of the form x^n, where n is any real number except -1, the antiderivative is (1/(n+1))x^(n+1). Therefore, the antiderivative of f(x) = x^7 is F(x) = (1/8)x^8 + C, where C is the constant of integration.

ii) The antiderivative of f(x) = x^(-3) can be found using the power rule again. Since n = -3 in this case, the antiderivative is (1/(-3+1))x^(-3+1) = -(1/2)x^(-2). Thus, the antiderivative of f(x) = x^(-3) is F(x) = -(1/2)x^(-2) + C.

iii) The function f(x) = 3x^2 + 3x can be easily integrated term by term. The antiderivative of each term is found using the power rule. The antiderivative of 3x^2 is x^3, and the antiderivative of 3x is (3/2)x^2. Therefore, the antiderivative of f(x) = 3x^2 + 3x is F(x) = x^3 + (3/2)x^2 + C, where C is the constant of integration.