antiderivatives and indefinite integrals\nfind the particular antiderivative of the following derivative…

antiderivatives and indefinite integrals\nfind the particular antiderivative of the following derivative that satisfies the given condition.\nc(x)=5x^2 - 4x; c(0)=1,000\nc(x)=□

antiderivatives and indefinite integrals\nfind the particular antiderivative of the following derivative that satisfies the given condition.\nc(x)=5x^2 - 4x; c(0)=1,000\nc(x)=□

Answer

Answer:

$\frac{5}{3}x^{3}-2x^{2}+1000$

Explanation:

Step1: Integrate the derivative

$C(x)=\int(5x^{2}-4x)dx=\int5x^{2}dx-\int4xdx$

Step2: Apply power - rule for integration

$\int5x^{2}dx = 5\times\frac{x^{2 + 1}}{2+1}=\frac{5}{3}x^{3}$, $\int4xdx=4\times\frac{x^{1+1}}{1 + 1}=2x^{2}$, so $C(x)=\frac{5}{3}x^{3}-2x^{2}+C$

Step3: Use the given condition

Substitute $x = 0$ and $C(0)=1000$ into $C(x)$. We get $C(0)=\frac{5}{3}(0)^{3}-2(0)^{2}+C=1000$, so $C = 1000$. Then $C(x)=\frac{5}{3}x^{3}-2x^{2}+1000$