Solutions to Problem Set 1

Math 211-03

8-31-2017

[Integration and differentiation review]

1. Compute $\displaystyle \der {}
   x \left(8 \sqrt{x} - 5 x + \dfrac{3}{x} - 10\right)$ .

$$\der {} x \left(8 \sqrt{x} - 5 x + \dfrac{3}{x} - 10\right) = 4 x^{-1/2} - 5 - \dfrac{3}{x^2}.\quad\halmos$$


2. Compute $\displaystyle \der {}
   x (e^{3 x} + \tan x)(\cos x - \csc x)$ .

$$\der {} x (e^{3 x} + \tan x)(\cos x - \csc x) = (e^{3 x} + \tan x)(-\sin x + \csc x \cot x) + (\cos x - \csc x)[3 e^{3 x} + (\sec x)^2].\quad\halmos$$


3. Compute $\displaystyle \der {}
   x \dfrac{5 \ln x}{8 x + \tan^{-1} x}$ .

$$\der {} x \dfrac{5 \ln x}{8 x + \tan^{-1} x} = \dfrac{(8 x + \tan^{-1} x)\left(\dfrac{5}{x}\right) - (5 \ln x)\left(8 + \dfrac{1}{x^2 + 1}\right)}{(8 x + \tan^{-1} x)^2}.\quad\halmos$$


4. Compute $\displaystyle \int
   \left(12 \sqrt{x} + \dfrac{6}{x^3} - \dfrac{13}{x}\right)\,dx$ .

$$\int \left(12 \sqrt{x} + \dfrac{6}{x^3} - \dfrac{13}{x}\right)\,dx = 8 x^{3/2} - \dfrac{3}{x^2} - 13 \ln |x| + c.\quad\halmos$$


5. Compute $\displaystyle \int
   \dfrac{1}{\sqrt{9 - x^2}}\,dx$ .

$$\int \dfrac{1}{\sqrt{9 - x^2}}\,dx = \sin^{-1} \dfrac{x}{3} + c.\quad\halmos$$


6. Compute $\displaystyle \int
   (e^{3 x} + \cos 4 x)\,dx$ .

$$\int (e^{3 x} + \cos 4 x)\,dx = \dfrac{1}{3}e^{3 x} + \dfrac{1}{4}\sin 4 x + c.\quad\halmos$$


7. Compute $\displaystyle \int
   (x^2 - 1)(x + 2)\,dx$ .

$$\int (x^2 - 1)(x + 2)\,dx = \int (x^3 + 2 x^2 - x - 2)\,dx = \dfrac{1}{4} x^4 + \dfrac{2}{3} x^3 - \dfrac{1}{2} x^2 - 2 x + c. \quad\halmos$$


8. Compute $\displaystyle \int
   \dfrac{(e^x + 1)^2}{e^{3 x}}\,dx$ .

$$\int \dfrac{(e^x + 1)^2}{e^{3 x}}\,dx = \int \dfrac{e^{2 x} + 2 e^x + 1}{e^{3 x}}\,dx = \int \left(\dfrac{e^{2 x}}{e^{3 x}} + \dfrac{2 e^x}{e^{3 x}} + \dfrac{1}{e^{3 x}}\right)\,dx = \int \left(e^{-x} + 2 e^{-2 x} + e^{-3 x}\right)\,dx =$$

$$-e^{-x} - e^{-2 x} - \dfrac{1}{3} e^{-3 x} + c.\quad\halmos$$


9. Compute $\displaystyle \int
   \dfrac{x - 5}{\sqrt{x + 1}}\,dx$ .

$$\int \dfrac{x - 5}{\sqrt{x + 1}}\,dx = \int \dfrac{(u - 1) - 5}{\sqrt{u}}\,du = \int \dfrac{u - 6}{\sqrt{u}}\,du = \int \left(\dfrac{u}{\sqrt{u}} - \dfrac{6}{\sqrt{u}}\right)\,du =$$

$$\left[u = x + 1, \quad du = dx; \quad x = u - 1\right]$$

$$\int \left(\sqrt{u} - 6 u^{-1/2}\right)\,du = \dfrac{2}{3} u^{3/2} - 12 u^{1/2} + c = \dfrac{2}{3} (x + 1)^{3/2} - 12 (x + 1)^{1/2} + c.\quad\halmos$$


10. Compute $\displaystyle \int
   \left((\tan x)^3 + 7\tan x\right)(\sec x)^2\,dx$ .

$$\int \left((\tan x)^3 + 7\tan x\right)(\sec x)^2\,dx = \int (u^3 + 7u)(\sec x)^2\cdot \dfrac{du}{(\sec x)^2} = \int (u^3 + 7u)\,du =$$

$$\left[u = \tan x, \quad du = (\sec x)^2\,dx, \quad dx = \dfrac{du}{(\sec x)^2}\right]$$

$$\dfrac{1}{4}u^4 + \dfrac{7}{2}u^2 + c = \dfrac{1}{4}(\tan x)^4 + \dfrac{7}{2}(\tan x)^2 + c. \quad\halmos$$


11. Compute $\displaystyle \int
   \dfrac{e^{1/x^2}}{x^3}\,dx$ .

$$\int \dfrac{e^{1/x^2}}{x^3}\,dx = \int \dfrac{e^u}{x^3}\cdot \left(-\dfrac{x^3}{2}\right)\,du = -\dfrac{1}{2} \int e^u\,du = -\dfrac{1}{2}e^u + c = -\dfrac{1}{2}e^{1/x^2} + c.$$

$$\left[u = \dfrac{1}{x^2}, \quad du = -\dfrac{2}{x^3}\,dx, \quad dx = -\dfrac{x^3}{2}\,du\right]\quad\halmos$$


12. Compute $\displaystyle \int
   \dfrac{x + 5}{\sqrt{x^2 + 10 x + 5}}\,dx$ .

$$\int \dfrac{x + 5}{\sqrt{x^2 + 10 x + 5}}\,dx = \int \dfrac{x + 5}{\sqrt{u}}\cdot \dfrac{du}{2(x + 5)} = \dfrac{1}{2} \int u^{-1/2}\,du = \dfrac{1}{2} \cdot 2 u^{1/2} + c = \sqrt{x^2 + 10 x + 5} + c.$$

$$\left[u = x^2 + 10 x + 5, \quad du = (2 x + 10)\,dx = 2(x + 5)\,dx, \quad dx = \dfrac{du}{2(x + 5)}\right]\quad\halmos$$


13. Compute $\displaystyle \int
   \dfrac{dx}{x (\ln x + 3)^5}$ .

$$\int \dfrac{dx}{x (\ln x + 3)^5} = \int \dfrac{x\,du}{xu^5} = \int u^{-5}\,du = -\dfrac{1}{4} u^{-4} + c = -\dfrac{1}{4 (\ln x + 3)^4} + c.$$

$$\left[u = \ln x + 3, \quad du = \dfrac{dx}{x}, \quad dx = x\,du\right]\quad\halmos$$


14. Compute $\displaystyle \int
   \dfrac{1}{\cos x (\sin x)^2}\,dx$ .

$$\int \dfrac{1}{\cos x (\sin x)^2}\,dx = \int \dfrac{(\cos x)^2 + (\sin x)^2}{\cos x (\sin x)^2}\,dx = \int \left(\dfrac{(\cos x)^2}{\cos x (\sin x)^2} + \dfrac{(\sin x)^2}{\cos x (\sin x)^2}\right)\,dx =$$

$$\int \left(\dfrac{\cos x}{(\sin x)^2} + \dfrac{1}{\cos x}\right)\,dx = \int \left(\dfrac{1}{\sin x} \cdot \dfrac{\cos x}{\sin x} + \dfrac{1}{\cos x}\right)\,dx = \int (\csc x \cot x + \sec x)\,dx =$$

$$-\csc x + \ln |\sec x + \tan x| + c.\quad\halmos$$


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