Calculus for Economics, Exercises (Jan 09, 2018)

I Find the value of the following integrals.

(1) $\int^{\frac {\pi}2}_0 t\sin tdt$, (2) $\int^1_{-1} \frac 1{\sqrt{x+2}}dx$, (3) $\int^1_0 x(x-1)^3dx$, (4) $\int^6_{0} \left(\frac 13x-1\right)^4dx$,

(5) $\int^{-1}_{-3} \frac 1{(2x+1)^3}dx$, (6) $\int^{1}_{0} (x+1)e^xdx$, (7) $\int^{1}_{-1} (x+1)^3(x-1)dx$ （部分積分で）

II Find the value of the following integrals.

(1) $\int^2_{-1}\frac x{\sqrt{3-x}}dx$, (2) $\int^1_{0}\frac {x-1}{(2-x)^2}dx$, (3) $\int^2_{1}x{\sqrt{2-x}}dx$, (4) $\int^6_{0}\left(\frac x3-1\right)^4dx$,

(5) $\int^2_{1}\frac {e^x}{e^x+1}dx$, (6) $\int^2_{1}\frac {e^x}{(e^x+1)^2}dx$, (7) $\int^e_{1}\frac {(\log x)^2}{x}dx$, (8) $\int^1_{0}\sqrt{3-2x}dx$

III Find the value of the following integrals.

(1) $\int^8_1\sqrt[3]xdx$ (2) $\int^2_1\frac 1{y^3}dy$ (3) $\int^1_0x\sqrt xdx$ (4) $\int^{\frac {\pi}2}_0t\cos tdt$ (5) $\int^e_1t\log tdt$

(6) $\int^1_{-1}\frac {dx}{\sqrt {x+2}}$ (7) $\int^e_1\left(\log x\right)^2dx$ (8) $\int^1_0te^{-t^2}dt$ (9) $\int^{\frac 12}_0t\sqrt{1-t^2}dt$ (10) $\int^1_0x(x-1)^3dx$

(11) $\int^6_0\left(\frac 13 x-1\right)^4dx$ (12) $\int^{\frac {\pi}4}_0\frac {\sin x}{1+\cos x}dx$ (13) $\int^{\frac {\pi}2}_0\sin^3x\cos xdx$ (14) $\int^2_1\log(x+1)dx$ (15) $\int^e_1\left(2x-1\right)\log xdx$

(16) $\int^{-1}_{-3}\frac 1{(2x+1)^3}dx$ (17) $\int^1_0\frac {x-1}{(x-2)^2}dx$ (18) $\int^3_2\frac 1{x^2-1}dx$ (19) $\int^2_1x\log(x+1)dx$ (20) $\int^{\frac {\pi}2}_0\cos 2xdx$