Solutions to Problem Set 2

Math 211-03

9-5-2017

[Integration by parts]

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

$$\matrix{ & \displaystyle \der {} x & \displaystyle \int\,dx \cr \noalign{\vskip2pt} + & x^2 & e^{-3 x} \cr \noalign{\vskip2pt} - & 2 x & -\dfrac{1}{3} e^{-3 x} \cr \noalign{\vskip2pt} + & 2 & \dfrac{1}{9} e^{-3 x} \cr \noalign{\vskip2pt} - & 0 & -\dfrac{1}{27} e^{-3 x} \cr}$$

$$\int x^2 e^{-3 x} \,dx = -\dfrac{1}{3} x^2 e^{-3 x} - \dfrac{2}{9} x e^{-3 x} - \dfrac{2}{27} e^{-3 x} + c.\quad\halmos$$


2. Compute $\displaystyle \int x^2
   \cos 4 x\,dx$ .

$$\matrix{ & \displaystyle \der {} x & \displaystyle \int\,dx \cr \noalign{\vskip2pt} + & x^2 & \cos 4 x \cr \noalign{\vskip2pt} - & 2 x & \dfrac{1}{4} \sin 4 x \cr \noalign{\vskip2pt} + & 2 & -\dfrac{1}{16} \cos 4 x \cr \noalign{\vskip2pt} - & 0 & -\dfrac{1}{64} \sin 4 x \cr}$$

$$\int x^2 \cos 4 x\,dx = \dfrac{1}{4} x^2 \sin 4 x + \dfrac{1}{8} x \cos 4 x - \dfrac{1}{32} \sin 4 x + c.\quad\halmos$$


3. Compute $\displaystyle \int (3
   x + 2)^2 \sin x\,dx$ .

$$\matrix{ & \displaystyle \der {} x & \displaystyle \int\,dx \cr \noalign{\vskip2pt} + & (3 x + 2)^2 & \sin x \cr \noalign{\vskip2pt} - & 6(3 x + 2) & -\cos x \cr \noalign{\vskip2pt} + & 18 & -\sin x \cr \noalign{\vskip2pt} - & 0 & \cos x \cr}$$

$$\int (3 x + 2)^2 \sin x\,dx = -(3 x + 2)^2 \cos x + 6(3 x + 2) \sin x + 18 \cos x + c.\quad\halmos$$


4. Compute $\displaystyle \int (3
   x + 1) (\sec x)^2\,dx$ .

$$\matrix{ & \displaystyle \der {} x & \displaystyle \int\,dx \cr \noalign{\vskip2pt} + & 3 x + 1 & (\sec x)^2 \cr \noalign{\vskip2pt} - & 3 & \tan x \cr \noalign{\vskip2pt} + & 0 & \ln |\sec x| \cr}$$

$$\int (3 x + 1) (\sec x)^2\,dx = (3 x + 1) \tan x - 3 \ln |\sec x| + c.\quad\halmos$$


5. Compute $\displaystyle \int
   x^6 \ln x\,dx$ .

$$\matrix{ & \displaystyle \der {} x & \displaystyle \int\,dx \cr \noalign{\vskip2pt} + & \ln x & x^2 \cr \noalign{\vskip2pt} - & \dfrac{1}{x} & \dfrac{1}{7} x^7 \cr}$$

$$\int x^2 \ln x\,dx = \dfrac{1}{7} x^7 \ln x - \int \dfrac{1}{x} \cdot \dfrac{1}{7} x^7\,dx = \dfrac{1}{7} x^7 \ln x - \dfrac{1}{7} \int x^6\,dx = \dfrac{1}{7} x^7 \ln x - \dfrac{1}{49} x^7 + c.\quad\halmos$$


6. Compute $\displaystyle \int
   \tan^{-1} x\,dx$ .

$$\matrix{ & \displaystyle \der {} x & \displaystyle \int\,dx \cr \noalign{\vskip2pt} + & \tan^{-1} x & 1 \cr \noalign{\vskip2pt} - & \dfrac{1}{x^2 + 1} & x \cr}$$

$$\int \tan^{-1} x\,dx = x \tan^{-1} x - \int \dfrac{x}{x^2 + 1}\,dx = x \tan^{-1} x - \int \dfrac{x}{u} \cdot \dfrac{du}{2 x} = x \tan^{-1} x - \dfrac{1}{2} \int \dfrac{1}{u}\,du =$$

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

$$x \tan^{-1} x - \dfrac{1}{2} \ln |u| + c = x \tan^{-1} x - \dfrac{1}{2} \ln |x^2 + 1| + c.\quad\halmos$$


7. Compute $\displaystyle \int (3
   x + 1) (x - 3)^{50}\,dx$ .

$$\matrix{ & \displaystyle \der {} x & \displaystyle \int\,dx \cr \noalign{\vskip2pt} + & 3 x + 1 & (x - 3)^{50} \cr \noalign{\vskip2pt} - & 3 & \dfrac{1}{51} (x - 3)^{51} \cr \noalign{\vskip2pt} + & 0 & \dfrac{1}{2652} (x - 3)^{52} \cr}$$

$$\int (3 x + 1) (x - 3)^{50}\,dx = \dfrac{1}{51} (3 x + 1) (x - 3)^{51} - \dfrac{1}{884} (x - 3)^{52} + c.\quad\halmos$$

You can also do this problem by substitution: Let $u = x - 3$ .


8. Compute $\displaystyle \int
   e^{3 x} \sin 5 x\,dx$ .

$$\matrix{ & \displaystyle \der {} x & \displaystyle \int\,dx \cr \noalign{\vskip2pt} + & e^{3 x} & \sin 5 x \cr \noalign{\vskip2pt} - & 3 e^{3 x} & -\dfrac{1}{5} \cos 5 x \cr \noalign{\vskip2pt} + & 9 e^{3 x} & -\dfrac{1}{25} \sin 5 x \cr}$$

$$\eqalign{ \int e^{3 x} \sin 5 x\,dx & = -\dfrac{1}{5} e^{3 x} \cos 5 x + \dfrac{3}{25} e^{3 x} \sin 5 x - \dfrac{9}{25} \int e^{3 x} \sin 5 x\,dx \cr \dfrac{34}{25} \int e^{3 x} \sin 5 x\,dx & = -\dfrac{1}{5} e^{3 x} \cos 5 x + \dfrac{3}{25} e^{3 x} \sin 5 x \cr \int e^{3 x} \sin 5 x\,dx & = -\dfrac{5}{34} e^{3 x} \cos 5 x + \dfrac{3}{34} e^{3 x} \sin 5 x + c \quad\halmos \cr}$$


9. Compute $\displaystyle \int
   \sin 8 x \sin 2 x\,dx$ .

$$\matrix{ & \displaystyle \der {} x & \displaystyle \int\,dx \cr \noalign{\vskip2pt} + & \sin 8 x & \sin 2 x \cr \noalign{\vskip2pt} - & 8 \cos 8 x & -\dfrac{1}{2} \cos 2 x \cr \noalign{\vskip2pt} + & -64 \sin 8 x & -\dfrac{1}{4} \sin 2 x \cr}$$

$$\int \sin 8 x \sin 2 x\,dx = -\dfrac{1}{2} \sin 8 x \cos 2 x + 2 \cos 8 x \sin 2 x + 16 \int \sin 8 x \sin 2 x\,dx$$

$$-15 \int \sin 8 x \sin 2 x\,dx = -\dfrac{1}{2} \sin 8 x \cos 2 x + 2 \cos 8 x \sin 2 x$$

$$\int \sin 8 x \sin 2 x\,dx = \dfrac{1}{30} \sin 8 x \cos 2 x - \dfrac{2}{15} \cos 8 x \sin 2 x + c.\quad\halmos$$

You can also do this problem using the trig identity

$$\sin 8 x \sin 2 x = \dfrac{1}{2} \left(\cos 6 x - \cos 10 x\right).$$


10. Compute $\displaystyle
   \int_0^{\pi/2} x^2 \sin 2 x\,dx$ .

$$\matrix{ & \displaystyle \der {} x & \displaystyle \int\,dx \cr \noalign{\vskip2pt} + & x^2 & \sin 2 x \cr \noalign{\vskip2pt} - & 2 x & -\dfrac{1}{2} \cos 2 x \cr \noalign{\vskip2pt} + & 2 & -\dfrac{1}{4} \sin 2 x \cr \noalign{\vskip2pt} - & 0 & \dfrac{1}{8} \cos 2 x \cr}$$

$$\int_0^{\pi/2} x^2 \sin 2 x\,dx = \left[-\dfrac{1}{2} x^2 \cos 2 x + \dfrac{1}{2} x \sin 2 x + \dfrac{1}{4} \cos 2 x\right]_0^{\pi/2} =$$

$$\left(\dfrac{\pi^2}{8} + 0 - \dfrac{1}{4}\right) - \left(0 + 0 + \dfrac{1}{4}\right) = \dfrac{\pi^2}{8} - \dfrac{1}{2} = 0.73370 \ldots .\quad\halmos$$


11. Compute $\displaystyle
   \int_0^1 (x + 7) \sqrt{x + 5}\,dx$ .

$$\matrix{ & \displaystyle \der {} x & \displaystyle \int\,dx \cr \noalign{\vskip2pt} + & x + 7 & \sqrt{x + 5} \cr \noalign{\vskip2pt} - & 1 & \dfrac{2}{3} (x + 5)^{3/2} \cr \noalign{\vskip2pt} + & 0 & \dfrac{4}{15} (x + 5)^{5/2} \cr}$$

$$\int_0^1 (x + 7) \sqrt{x + 5}\,dx = \left[\dfrac{2}{3} (x + 7) (x + 5)^{3/2} - \dfrac{4}{15} (x + 5)^{5/2}\right]_0^1 =$$

$$\left(\dfrac{16}{3} 6^{3/2} - \dfrac{4}{15} 6^{5/2}\right) - \left(\dfrac{14}{3} 5^{3/2} - \dfrac{4}{15} 5^{5/2}\right) = 17.60077 \ldots .\quad\halmos$$


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