Antiderivatives and Indefinite IntegralsDefinitionThe function \( f(x)=3x^2 \) has many antiderivatives: \( F_1(x)=x^3 \), \( F_2(x)=x^317 \), \( F_3(x)=x^3+41 \), etc. They all differ by a constant factor. Consider the function \( f(x)=\cos x \). For each real number \( C \), the function \( F_C(x)=\sin x+C \) is an antiderivative of \( f \). We write \[ \int \cos x\,dx=\sin x+C.\] Main properties of indefinite integralsThe following theorem is easy to prove using the main properties of derivatives. We can use the previous theorem to find antiderivatives of polynomials. For example, if \( P(x)=x^32x^2+11x+4 \), then \[ \int P(x)\,dx=\int x^3\,dx2\int x^2\,dx+11\int x\,dx+4\int 1\,dx= \frac14x^32\frac{x^3}3+11\frac{x^2}2+4x+C,\] where \( C \) could be any real number.

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