4.1 · Antiderivatives and initial conditions

An indefinite integral describes a family

If $\displaystyle F'(x)=f(x)$, then $\displaystyle F$ is an antiderivative of $\displaystyle f$. Adding any constant leaves the derivative unchanged, so an indefinite integral includes $\displaystyle +C$.

For $\displaystyle n\ne-1$, increase the power by one and divide by that new power. Verify the result by differentiating rather than relying on its appearance.

$$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\qquad n\ne-1$$

Recognize the logarithmic exception

The power rule cannot divide by zero when the integrand is $\displaystyle x^{-1}$. Instead, $\displaystyle \int (1/x)\,dx=\ln|x|+C$ on intervals excluding zero. The absolute value allows either positive or negative input intervals.

Other basic pairs include $\displaystyle \int e^x\,dx=e^x+C$, $\displaystyle \int\cos x\,dx=\sin x+C$, and $\displaystyle \int\sin x\,dx=-\cos x+C$. Constants multiply the whole antiderivative.

Fit the initial condition after integrating

If $\displaystyle y'=2x$ and $\displaystyle y(1)=4$, integration gives $\displaystyle y=x^2+C$. Substituting the point gives $\displaystyle 4=1+C$, hence $\displaystyle C=3$. The initial condition selects one member of the family.

If acceleration is given and position is requested, integrate twice, with a different constant at each stage. Velocity data determines the first constant; position data determines the second.