1. The problem is to understand and apply the method of $u$-substitution in integration. 2. $u$-substitution is used to simplify integrals by substituting a part of the integral with a new variable $u$. 3. The general formula is: If $u = g(x)$, then $\frac{du}{dx} = g'(x)$ or $du = g'(x) dx$. 4. Replace the integral in terms of $x$ with an integral in terms of $u$ using $du$. 5. Solve the integral in $u$, then substitute back $u = g(x)$ to get the answer in terms of $x$. 6. Example: To integrate $\int 2x \cos(x^2) dx$, let $u = x^2$, then $du = 2x dx$. 7. The integral becomes $\int \cos(u) du$, which integrates to $\sin(u) + C$. 8. Substitute back $u = x^2$ to get the final answer: $\sin(x^2) + C$. This method helps simplify complex integrals by changing variables.