Exemplos de substituição
Exemplo 2. $\int_0^{\pi/2}\cos x\,dx$. Direto: $[\sin x]_0^{\pi/2}=1$.
Exemplo 3. $\int_0^1\frac{x}{\sqrt{1+x^2}}\,dx$. Com $u=1+x^2$: $du=2x\,dx$, limites $1\to 2$:
$$=\frac{1}{2}\int_1^2 u^{-1/2}\,du=\frac{1}{2}[2u^{1/2}]_1^2=\sqrt{2}-1\approx 0{,}4142.$$
Exemplo 4. $\int_1^e\frac{(\ln x)^2}{x}\,dx$. Com $u=\ln x$: $du=dx/x$, limites $0\to 1$:
$$=\int_0^1 u^2\,du=\frac{1}{3}.$$