[tex]\boxed{\int\limits_0^1{\tan{x}\sin{x}dx}}[/tex]
Primeiramente, podemos realizar algumas manipulações trigonométricas:
[tex]\int\limits_0^1{\dfrac{\sin{x}}{\cos{x}}\sin{x}dx}=\int\limits_0^1{\sec{x}\sin^2{x}dx}=[/tex]
[tex]\int\limits_0^1{\sec{x}(1-\cos^2{x})dx}=\int\limits_0^1{(\sec{x}-\cos{x})dx}=[/tex]
[tex]\bigg(\int{\sec{x}dx}-\int{\cos{x}dx}\bigg)\bigg|^1_0[/tex]
As integrais de [tex]\sec{x}[/tex] e [tex]\cos{x}[/tex] já são conhecidas:
[tex]\bigg(\int{\sec{x}dx}-\int{\cos{x}dx}\bigg)\bigg|^1_0=[/tex]
[tex]\bigg(\ln{(|\tan{x}+\sec{x}|)}-\sin{x}\bigg)\bigg|^1_0=[/tex]
[tex]\ln{(|\tan{1}+\sec{1}|)}-\sin{1}-\ln{|\tan{0}+\sec{0}|}+\sin{0}=[/tex]
[tex]\boxed{\ln{(|\tan{1}+\sec{1}|)}-\sin{1}}[/tex]
