4.4 · Substitution and inverse-trigonometric integrals

Find an inside function and its differential

For $\displaystyle \int2x\cos(x^2)\,dx$, set $\displaystyle u=x^2$ so $\displaystyle du=2x\,dx$. The integral becomes $\displaystyle \int\cos u\,du=\sin u+C$, then return to $\displaystyle x$. The substitution must account for every factor.

A constant discrepancy can be adjusted, but a missing variable factor cannot simply be discarded. Verify the resulting antiderivative by differentiating it.

$$\int f(g(x))g'(x)\,dx=F(g(x))+C$$

Keep definite-integral bounds consistent

For a definite integral, transform each original bound through $\displaystyle u=g(x)$ and integrate in $\displaystyle u$, or find an antiderivative in $\displaystyle x$ and use the original bounds. Do not mix $\displaystyle u$ expressions with $\displaystyle x$ bounds.

If $\displaystyle u=x^2+1$ on $\displaystyle [0,1]$, the new bounds are $\displaystyle 1$ and $\displaystyle 2$. A transformed bound depends on the substitution, not on the differential alone.

Recognize logarithmic and inverse-trigonometric patterns

The integral of $\displaystyle g'(x)/g(x)$ is $\displaystyle \ln|g(x)|+C$ on suitable intervals. The pattern $\displaystyle 1/(1+x^2)$ integrates to $\displaystyle \arctan x+C$, and $\displaystyle 1/\sqrt{1-x^2}$ integrates to $\displaystyle \arcsin x+C$ on $\displaystyle |x|<1$.

For a scaled arctangent pattern, account for the scale: $\displaystyle \int dx/(a^2+x^2)=(1/a)\arctan(x/a)+C$ for $\displaystyle a>0$. The chain-rule check confirms the coefficient.