91Ó°ÊÓ

Solve the given problems by integration.The force \(F\) (in \(\mathrm{N}\) ) applied by a stamping machine in making a certain computer part is \(F=4 x /\left(x^{2}+3 x+2\right),\) where \(x\) is the distance (in \(\mathrm{cm}\) ) through which the force acts. Find the work done by the force from \(x=0\) to \(x=0.500 \mathrm{cm}\).

Solve the given problems by integration. If the current \(i\) (in \(\mathrm{A}\) ) in a certain electric circuit is given by \(i=110 \cos 377 t,\) find the expression for the voltage across a \(500-\mu \mathrm{F}\) capacitor as a function of time. The initial voltage is zero. Show that the voltage across the capacitor is \(90^{\circ}\) out of phase with the current.

Solve the given problems by integration. Find the first-quadrant area bounded by \(y=\frac{\ln (4 x+1)}{4 x+1}\) and \(x=5\).

Show that \(\int_{0}^{3} \frac{d x}{2 x+2}=\ln 2\).

Solve the given problems by integration. The vertical cross section of a highway culvert is defined by the region within the ellipse \(1.00 x^{2}+9.00 y^{2}=9.00,\) where dimensions are in meters. Find the area of the cross section of the culvert.

Solve the given problems by integration. Integrate \(\int \sqrt{\frac{1+x}{1-x}} d x\) by first multiplying the numerator and denominator of the fraction under the radical by \(1+x\).

In Exercises \(27-42,\) solve the given problems by integration. The general expression for the slope of a curve is \(d y / d x=x^{3} \sqrt{1+x^{2}} .\) Find the equation of the curve if it passes through the origin.

$$\text {Solve the given problems by integration.}$$ The general expression for the slope of a curve is \(d y / d x=x^{3} \sqrt{1+x^{2}} .\) Find the equation of the curve if it passes through the origin.

Solve the given problems by integration. $$\text { Show that } \int_{0}^{3} \frac{d x}{2 x+2}=\ln 2$$

Solve the given problems by integration. Under specified conditions, the time \(t\) (in min) required to form \(x\) grams of a substance during a chemical reaction is given by \(t=\int d x /[(4-x)(2-x)] .\) Find the equation relating \(t\) and \(x\) if \(x=0\) g when \(t=0\) min.