3.1 不定積分

テキスト99ページの積分公式を用います．

1.

(a) $${\int (2x -3)\; dx \underbrace{=}_{(1)} 2\cdot \frac{x^2}{2} - 3x + c = x^2 - 3x + c}$$

(b) $${\int 5x^4\; dx \underbrace{=}_{(1)} 5 \cdot \frac{x^5}{5} + c = x^5 + c}$$

(c) $${\int \sqrt[3]{x^2}\; dx \underbrace{=}_{(1)} \int x^{2/3}dx = \frac{3}{5}x^{5/3} + c }$$

(d) $${\int \frac{1}{\sqrt{x}}\; dx \underbrace{=}_{(1)} \int x^{-1/2}dx = 2x^{1/2} + c }$$

(e) $${\int \frac{\sin{x}}{\cos{x}}\; dx \underbrace{=}_{(6)} \int \tan{x}\; dx = \log{\vert\sec{x}\vert} + c }$$

(f) $${\int \frac{1}{\cos^{2}{x}}\; dx \underbrace{=}_{(7)} \int \sec^{2}{x} dx = \tan{x} + c }$$

(g) $${\int \sin^{2}{x}\; dx = \int \frac{1 - \cos{2x}}{2}\; dx = \frac{1}{2}(x + \frac{\sin{2x}}{2}) + c }$$

(h) $${\int \frac{1}{x^2 - 4}\; dx \underbrace{=}_{(9)} \int \frac{1}{x^2 - 2^2}\; dx = \frac{1}{4}\log{\vert\frac{x-2}{x+2}\vert} + c }$$

(i) $${\int \frac{1}{x^2 + 4}\; dx \underbrace{=}_{(10)} \int \frac{1}{x^2 + 2^2}\; dx = \frac{1}{2}\tan^{-1}{\frac{x}{2}} + c }$$