91Ó°ÊÓ

Solve the differential equation subject to the given conditions. $$ f^{\prime \prime}(x)=4 x-1: \quad f(2)=-2 ; \quad f(1)=3 $$

$$ D_{x} \int_{2}^{x^{4}} \frac{t}{\sqrt{t^{3}+2}} d t $$

Exer. \(53-56:\) Evaluate the integral by (a) the method of substitution and (b) expanding the integrand. In what way do the constants of integration differ? $$ \int(x+4)^{2} d x $$

Solve the differential equation subject to the given conditions. $$ f^{\prime \prime}(x)=6 x-4 ; \quad f^{\prime}(2)=5 ; \quad f(2)=4 $$

Exer. \(53-56:\) Evaluate the integral by (a) the method of substitution and (b) expanding the integrand. In what way do the constants of integration differ? $$ \int\left(x^{2}+4\right)^{2} x d x $$

Exer. \(53-56:\) Evaluate the integral by (a) the method of substitution and (b) expanding the integrand. In what way do the constants of integration differ? $$ \int \frac{(\sqrt{x}+3)^{2}}{\sqrt{x}} d x $$

Solve the differential equation subject to the given conditions. $$ \frac{d^{2} y}{d x^{2}}=3 \sin x-4 \cos x ; \quad y=7 \text { and } y^{\prime}=2 \text { if } x=0 $$

Exer. \(53-56:\) Evaluate the integral by (a) the method of substitution and (b) expanding the integrand. In what way do the constants of integration differ? $$ \int\left(1+\frac{1}{x}\right)^{2} \frac{1}{x^{2}} d x $$

Solve the differential equation subject to the given conditions. $$ \frac{d^{2} y}{d x^{2}}=2 \cos x-5 \sin x ; \quad y=2+6 \pi \text { and } y^{\prime}=3 \text { if } x=\pi $$

A charged particle is moving on a coordinate line in a magnetic field such that its velocity (in \(\mathrm{cm} / \mathrm{sec}\) ) at time \(t\) is given by \(v(t)=\frac{1}{2} \sin \left(3 t-\frac{1}{4} \pi\right)\). Show that the motion is simple harmonic (see page 223 ).