Today's calculation of Integral 366
Source: 1989 Oita University entrance examination/Medical
July 21, 2008
calculusintegrationtrigonometrycalculus computations
Problem Statement
(1) Determine the constant numbers such that the following equation is equality.
\ \ \ \ \ 4x \equal{} \{a(x \plus{} 1)^2 \plus{} b(x \plus{} 1) \plus{} c\}(x^2 \plus{} 1)^2 \plus{} \{(px \plus{} q)(x^2 \plus{} 1) \plus{} (rx \plus{} s)\}(x \plus{} 1)^3.
(2) Evaluate the following definite integrals.
(a) \displaystyle \int_0^1 \frac {dx}{x \plus{} 1}\ \ \ \ \ \ \ \ \ (b)\ \int_0^1 \frac {dx}{(x \plus{} 1)^2}\ \ \ \ \ (c)\ \int_0^1 \frac {dx}{(x \plus{} 1)^3}
(d) \displaystyle \int_0^1 \frac {x}{x^2 \plus{} 1}\ dx\ \ \ \ (e)\ \int_0^1 \frac {dx}{x^2 \plus{} 1}\ \ \ \ \ \ \ (f)\ \int_0^1 \frac {x}{(x^2 \plus{} 1)^2}\ dx
(g) \displaystyle \int_0^1 \frac {dx}{(x^2 \plus{} 1)^2}\ \ \ \ (h)\ \int_0^1 \frac {4x}{(x \plus{} 1)^3(x^2 \plus{} 1)^2}\ dx