Answer
$(a)$ infinite discontinuity
$(b)$ Infinite interval
$(c)$ Infinite interval
$(d)$ Infinite discontinuity
Work Step by Step
$(a)\;\; I=\int_{1}^{2}\frac{x}{x-1}dx$ _____(1)
Because integrand $\frac{x}{x-1}$ is discontinuous at $x=1$
That's why (1) is Type 2 improper integral.
$(b)\;\; \int_{0}^{\infty}\frac{1}{1+x^3}dx$ ____(2)
Because interval of integration is infinite
That's why (2) is Type (1) improper integral.
$(c)\;\; \int_{-\infty}^{\infty}x^2\:e^{-x^2}dx$ ____(3)
Because interval of integration is infinte
That's why (3) is Type (1) improper integral.
$(d)\;\; \int_{0}^{\frac{π}{4}}\cot xdx=\int_{0}^{\frac{π}{4}}\frac{\cos x}{\sin x}dx$ __(4)
Since $\sin (0)=0$
So integrand $\cot x$ is discontinuous at $x=0$
That's why (4) is Type (2) improper integral.
