Integrating f(ax+b) and by inspection Problem set A One question at a time

Find each integral.

1. (a) \(\int \sqrt{5x-2}\,dx\)

   • Integrate as normal, treating the inside as x (add 1 to the power and divide by the new power, 3/2): \(\frac{2}{3}(5x-2)^{\frac{3}{2}}\).
   • Divide by 5, the coefficient of x: \(\frac{2}{3}\div5=\frac{2}{15}\).
   • \(\int \sqrt{5x-2}\,dx=\frac{2}{15}(5x-2)^{\frac{3}{2}}+c\)

2. (b) \(\int \sqrt[3]{4x+3}\,dx\)

   • Integrate as normal, treating the inside as x (add 1 to the power and divide by the new power, 4/3): \(\frac{3}{4}(4x+3)^{\frac{4}{3}}\).
   • Divide by 4, the coefficient of x: \(\frac{3}{4}\div4=\frac{3}{16}\).
   • \(\int \sqrt[3]{4x+3}\,dx=\frac{3}{16}(4x+3)^{\frac{4}{3}}+c\)

Find \(\int 12(3x+1)^{-\frac{4}{3}}\,dx\).

• Integrate as normal, treating the inside as x (add 1 to the power and divide by the new power, -1/3): \(-36(3x+1)^{-\frac{1}{3}}\).
• Divide by 3, the coefficient of x: \(-36\div3=-12\).
• \(\int 12(3x+1)^{-\frac{4}{3}}\,dx=-12(3x+1)^{-\frac{1}{3}}+c\)

Find each integral.

1. (a) \(\int \cos(4x-1)\,dx\)

   • Integrate as normal, treating the inside as x (the integral of cos is sin): \(\sin(4x-1)\).
   • Divide by 4, the coefficient of x: \(1\div4=\frac{1}{4}\).
   • \(\int \cos(4x-1)\,dx=\frac{1}{4}\sin(4x-1)+c\)

2. (b) \(\int 5\cos(2x+3)\,dx\)

   • Integrate as normal, treating the inside as x (the integral of cos is sin): \(5\sin(2x+3)\).
   • Divide by 2, the coefficient of x: \(5\div2=\frac{5}{2}\).
   • \(\int 5\cos(2x+3)\,dx=\frac{5}{2}\sin(2x+3)+c\)

Find each integral.

1. (a) \(\int \sin(3x+2)\,dx\)

   • Integrate as normal, treating the inside as x (the integral of sin is minus cos): \(-\cos(3x+2)\).
   • Divide by 3, the coefficient of x: \(-1\div3=-\frac{1}{3}\).
   • \(\int \sin(3x+2)\,dx=-\frac{1}{3}\cos(3x+2)+c\)

2. (b) \(\int 4\sin(1-2x)\,dx\)

   • Integrate as normal, treating the inside as x (the integral of sin is minus cos): \(-4\cos(1-2x)\).
   • Divide by -2, the coefficient of x: \(-4\div(-2)=2\).
   • \(\int 4\sin(1-2x)\,dx=2\cos(1-2x)+c\)

Find each integral.

1. (a) \(\int 4e^{5x+1}\,dx\)

   • Integrate as normal, treating the inside as x (the integral of e to a power is itself): \(4e^{5x+1}\).
   • Divide by 5, the coefficient of x: \(4\div5=\frac{4}{5}\).
   • \(\int 4e^{5x+1}\,dx=\frac{4}{5}e^{5x+1}+c\)

2. (b) \(\int 10e^{1-2x}\,dx\)

   • Integrate as normal, treating the inside as x (the integral of e to a power is itself): \(10e^{1-2x}\).
   • Divide by -2, the coefficient of x: \(10\div(-2)=-5\).
   • \(\int 10e^{1-2x}\,dx=-5e^{1-2x}+c\)

Find each integral.

1. (a) \(\int \frac{6}{2x+5}\,dx\)

   • Integrate as normal, treating the inside as x (the integral of 1/u is ln|u|): \(6\ln|2x+5|\).
   • Divide by 2, the coefficient of x: \(6\div2=3\).
   • \(\int \frac{6}{2x+5}\,dx=3\ln|2x+5|+c\)

2. (b) \(\int \frac{3}{4-3x}\,dx\)

   • Integrate as normal, treating the inside as x (the integral of 1/u is ln|u|): \(3\ln|4-3x|\).
   • Divide by -3, the coefficient of x: \(3\div(-3)=-1\).
   • \(\int \frac{3}{4-3x}\,dx=-\ln|4-3x|+c\)

Find \(\int 12x^{3}(x^4-2)^{3}\,dx\).

• Spot the pair: \(g(x)=x^4-2\) and \(g'(x)=4x^{3}\).
• The integrand is \(3g'(x)[g(x)]^{3}\), so the integral is \(3\times\frac{(x^4-2)^{4}}{4}\).
• \(\int 12x^{3}(x^4-2)^{3}\,dx=\frac{3}{4}(x^4-2)^{4}+c\)
• Check: differentiate the answer and you get the integrand back.

Find each integral.

1. (a) \(\int 5\cos x\sin^{4}x\,dx\)

   • \(\cos x\) is the derivative of \(\sin x\): take \(g(x)=\sin x\).
   • The integral is \(5\times\frac{\sin^{5}x}{5}\).
   • \(\int 5\cos x\sin^{4}x\,dx=\sin^{5}x+c\)
   • Check: differentiate the answer and you get the integrand back.

2. (b) \(\int \sin x\cos^{2}x\,dx\)

   • \(\sin x\) is minus the derivative of \(\cos x\): take \(g(x)=\cos x\), \(g'(x)=-\sin x\).
   • Allow for the minus sign: the integral is \(-\frac{\cos^{3}x}{3}\).
   • \(\int \sin x\cos^{2}x\,dx=-\frac{1}{3}\cos^{3}x+c\)
   • Check: differentiate the answer and you get the integrand back.

Find each integral.

1. (a) \(\int \frac{4x}{x^2+7}\,dx\)

   • Spot the pair: \(g(x)=x^2+7\) and \(g'(x)=2x\).
   • The top is \(2g'(x)\), so the integral is \(2\ln|g(x)|\).
   • \(\int \frac{4x}{x^2+7}\,dx=2\ln|x^2+7|+c\)
   • Check: differentiate the answer and you get the integrand back.

2. (b) \(\int \frac{3x^{2}}{x^3+2}\,dx\)

   • Spot the pair: \(g(x)=x^3+2\) and \(g'(x)=3x^{2}\).
   • The top is exactly \(g'(x)\), so the integral is \(\ln|g(x)|\).
   • \(\int \frac{3x^{2}}{x^3+2}\,dx=\ln|x^3+2|+c\)
   • Check: differentiate the answer and you get the integrand back.

Find \(\int 8xe^{x^2-3}\,dx\).

• Spot the pair: \(g(x)=x^2-3\) and \(g'(x)=2x\).
• The integrand is \(4g'(x)e^{g(x)}\), so the integral is \(4e^{g(x)}\).
• \(\int 8xe^{x^2-3}\,dx=4e^{x^2-3}+c\)
• Check: differentiate the answer and you get the integrand back.

Find \(\int \frac{12}{\cos^2(3x-2)}\,dx\).

• \(\int\frac{1}{\cos^2 u}\,du=\tan u\)
• Divide by 3, the coefficient of x: \(12\div 3=4\)
• \(\int\frac{12}{\cos^2(3x-2)}\,dx=4\tan(3x-2)+c\)
• Check: differentiate and you get the integrand back.

A trolley slows down on a rough floor. Its velocity is \(\frac{ds}{dt}=40e^{-0.5t}\) cm/s, where \(s\) cm is the distance from the start and \(t\) is the time in seconds. When \(t=0\), \(s=0\).

1. (a) Find \(s\) in terms of \(t\).

   • \(s=\int 40e^{-0.5t}\,dt=-80e^{-0.5t}+c\): divide by \(-0.5\), the coefficient of t.
   • When \(t=0\), \(s=0\): \(0=-80+c\), so \(c=80\).

2. (b) Find \(s\) when \(t=4\), to 3 significant figures.

   • \(s=-80e^{-0.5\times 4}+80=69.2\) cm

The number of bacteria in a dish, \(N\), grows at the rate \(\frac{dN}{dt}=\frac{600}{2t+5}\) per hour, where \(t\) is the time in hours. When \(t=0\), \(N=1000\).

1. (a) Find \(N\) in terms of \(t\).

   • \(N=\int\frac{600}{2t+5}\,dt=300\ln(2t+5)+c\): divide by 2, the coefficient of t.
   • When \(t=0\), \(N=1000\): \(1000=300\ln 5+c\), so \(c=1000-300\ln 5\).

2. (b) Find \(N\) when \(t=10\), to 3 significant figures.

   • \(N=300\ln(2\times 10+5)+1000-300\ln 5=1480\) bacteria

Find \(\int x\sqrt{x^2+4}\,dx\).

• Spot the pair: \(g(x)=x^2+4\) and \(g'(x)=2x\).
• The integrand is \(\frac{1}{2}g'(x)[g(x)]^{\frac{1}{2}}\), so the integral is \(\frac{1}{2}\times\frac{(x^2+4)^{\frac{3}{2}}}{\frac{3}{2}}\).
• \(\int x\sqrt{x^2+4}\,dx=\frac{1}{3}(x^2+4)^{\frac{3}{2}}+c\)
• Check: differentiate the answer and you get the integrand back.

Given that \(f'(x)=\frac{8\cos x}{2+\sin x}\) and \(f(\frac{\pi}{2})=10\), find \(f(x)\), with the constant to 3 significant figures.

• The top is a multiple of the derivative of the bottom: \(g=2+\sin x\), \(g'=\cos x\).
• \(f(x)=8\ln(2+\sin x)+c\). The bottom is always positive, so no modulus is needed.
• Put in the point: \(10=8\ln(3)+c\), so \(c=10-8\ln 3\).
• \(c=1.21\), so \(f(x)=8\ln(2+\sin x)+1.21\)