\(\text{Let the given integral be I -}\)

\(I=\int_1^3(y^2+y)[\int_1^2 dx]dy\)

\(\text{Integrating with respect to x,}\)

\(I=\int_1^3(y^2+y)[x]_1^2 dy\)

\(I=\int_1^3(y^2+y)(2-1)dy\)

\(I=\int_1^3(y^2+y)dy\)

\(\text{Integrating equation with respect to y,}\)

\(I=\int_1^3(y^2+y)dy\)

\(I=[\frac{y^3}{3}+\frac{y^2}{2}]_1^3\)

\(I=(\frac{3^3}{3}+\frac{3^2}{2})-(\frac{1^3}{3}+\frac{1^2}{2})\)

\(I=\frac{27}{3}+\frac{9}{2}-\frac{1}{3}-\frac{1}{2}\)

\(I=\frac{54+27-2-3}{6}\)

\(I=\frac{76}{6}\)

\(I=\frac{38}{3}\)

\(\text{Hence,}\)

\(\int_1^3\int_1^2 (y^2+y)dxdy=\frac{38}{3}\)

\(I=\int_1^3(y^2+y)[\int_1^2 dx]dy\)

\(\text{Integrating with respect to x,}\)

\(I=\int_1^3(y^2+y)[x]_1^2 dy\)

\(I=\int_1^3(y^2+y)(2-1)dy\)

\(I=\int_1^3(y^2+y)dy\)

\(\text{Integrating equation with respect to y,}\)

\(I=\int_1^3(y^2+y)dy\)

\(I=[\frac{y^3}{3}+\frac{y^2}{2}]_1^3\)

\(I=(\frac{3^3}{3}+\frac{3^2}{2})-(\frac{1^3}{3}+\frac{1^2}{2})\)

\(I=\frac{27}{3}+\frac{9}{2}-\frac{1}{3}-\frac{1}{2}\)

\(I=\frac{54+27-2-3}{6}\)

\(I=\frac{76}{6}\)

\(I=\frac{38}{3}\)

\(\text{Hence,}\)

\(\int_1^3\int_1^2 (y^2+y)dxdy=\frac{38}{3}\)