Podrían ayudarme con la resolución de esta integral de línea?

Respuesta : 1.

2876 Explicación paso a paso : Integral de línea : <img src="https://tex.z-dn.net/?f=%5Cint%5Climits_C%20%7Bxy%7D%20%5C%2C%20dS%5C%20%5C%20%5C%20%5C%20%5C%20C%3Ax%3Dt%5E2%2C%20y%3D2t%3B%5C%20%5C%20%5C%20%5C%20%5C%200%5Cleq%20t%5Cleq%201" /> Primero se halla del diferencial de la integral dS : <img src="https://tex.z-dn.net/?f=dS%3D%5Csqrt%7B%28%5Cfrac%7Bdx%7D%7Bdt%7D%29%5E2%2B%28%5Cfrac%7Bdy%7D%7Bdt%7D%29%5E2%7Ddt%5C%5C%5C%5CdS%3D%5Csqrt%7B%28%5Cfrac%7Bd%28t%5E2%29%7D%7Bdt%7D%29%5E2%2B%28%5Cfrac%7Bd%282t%29%7D%7Bdt%7D%29%5E2%7Ddt%5C%5C%5C%5CdS%3D%5Csqrt%7B%282t%29%5E2%2B%282%29%5E2%7Ddt%5C%5C%5C%5CdS%3D%5Csqrt%7B4t%5E2%2B4%7Ddt%5C%5C%5C%5C%5Cboxed%7BdS%3D2%5Csqrt%7Bt%5E2%2B1%7Ddt%5C%5C%5C%5C%7D" /> Se sustituyen los valores de "x, y y dS" en la integral de línea : <img src="https://tex.z-dn.net/?f=%5Cint%5Climits%5E1_0%20%7B%28t%5E2%29%282t%29%7D%20%5C%2C2%5Csqrt%7Bt%5E2%2B1%7Ddt%5C%5C%5C%5C%5Cint%5Climits%5E1_0%20%7B4t%5E3%7D%20%5C%2C%5Csqrt%7Bt%5E2%2B1%7Ddt%5C%5C" /> Se realiza un cambio de variable : <img src="https://tex.z-dn.net/?f=u%3Dt%5E2%2B1%5C%5Ct%5E2%3Du-1%5C%5Cdu%3D2tdt%5C%5C%5C%5Cdt%3D%5Cdfrac%7Bdu%7D%7B2t%7D%5C%5C%5C%5Ct%3D0%5C%20%3D%3E%20u%3D1%5C%5Ct%3D1%5C%20%3D%3Eu%3D2%5C%5C%5C%5C%5C%5C%5Cint%5Climits%5E2_1%20%7B4%28u-1%29t%7D%20%5C%2C%5Csqrt%7Bu%7D%5Cdfrac%7Bdu%7D%7B2t%7D%5C%5C%5C%5C%5Cint%5Climits%5E2_1%20%7B2%28u-1%29%7D%20%5C%2C%5Csqrt%7Bu%7D%5C%20du%3D%5Cint%5Climits%5E2_1%20%7B2%28u-1%29%7Du%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%20%5C%2C%20du%3D2%5Cint%5Climits%5E2_1%20%7B%5Bu%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D-u%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%5D%20%5C%2C%20du" /> u = 1 \ \ t = 1 \ = >u = 2 \ \ \ \ \ \ \ int \ limits ^ 2_1 {4(u - 1)t} \ , \ sqrt{u} \ dfrac{du}{2t} \ \ \ \ \ int \ limits ^ 2_1 {2(u - 1)} \ , \ sqrt{u} \ du = \ int \ limits ^ 2_1 {2(u - 1)}u ^ { \ frac{1}{2}} \ , du = 2 \ int \ limits ^ 2_1 {[u ^ { \ frac{3}{2}} - u ^ { \ frac{1}{2}}] \ , du" alt = "u = t ^ 2 + 1 \ \ t ^ 2 = u - 1 \ \ du = 2tdt \ \ \ \ dt = \ dfrac{du}{2t} \ \ \ \ t = 0 \ = > u = 1 \ \ t = 1 \ = >u = 2 \ \ \ \ \ \ \ int \ limits ^ 2_1 {4(u - 1)t} \ , \ sqrt{u} \ dfrac{du}{2t} \ \ \ \ \ int \ limits ^ 2_1 {2(u - 1)} \ , \ sqrt{u} \ du = \ int \ limits ^ 2_1 {2(u - 1)}u ^ { \ frac{1}{2}} \ , du = 2 \ int \ limits ^ 2_1 {[u ^ { \ frac{3}{2}} - u ^ { \ frac{1}{2}}] \ , du" align = "absmiddle" class = "latex - formula"><img src="https://tex.z-dn.net/?f=2%280.4u%5E%7B%5Cfrac%7B5%7D%7B2%7D%7D-0.67u%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%29I%5E2_1%5C%5C%5C%5C%5Cboxed%7BI%3D1.2876%7D" />.