Each integral below yields to substitution

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation

Mathematical Methods for Physics, Ch Accumulation: integration and the techniques that make it possible

Problem

Each integral below yields to substitution, to integration by parts or a reduction formula, or to partial fractions, or to two of these in turn. For each one, say which method the form of the integrand calls for and why, evaluate the integral exactly, and check the result by differentiating your antiderivative back. ((a)) ∫0∞x3e−x2 dx\displaystyle\int_0^{\infty}x^3e^{-x^2}\,\mathrm{d}x ((b)) ∫01x2ex dx\displaystyle\int_0^{1}x^2e^{x}\,\mathrm{d}x ((c)) ∫123x+2x(x+1) dx\displaystyle\int_1^{2}\frac{3x+2}{x(x+1)}\,\mathrm{d}x

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