以下の4つの積分を計算します。 (1) $\int \frac{1}{\sqrt{x}} dx$ (2) $\int \left( \frac{1}{x} + \frac{1}{x^2} \right) dx$ (3) $\int \sqrt{x} dx$ (4) $\int \frac{1}{x^2 + 3} dx$

1. 問題の内容

以下の4つの積分を計算します。

(1) $\int \frac{1}{\sqrt{x}} dx$

(2) $\int \left( \frac{1}{x} + \frac{1}{x^2} \right) dx$

(3) $\int \sqrt{x} dx$

(4) $\int \frac{1}{x^2 + 3} dx$

2. 解き方の手順

(1) $\int \frac{1}{\sqrt{x}} dx = \int x^{-\frac{1}{2}} dx = \frac{x^{\frac{1}{2}}}{\frac{1}{2}} + C = 2\sqrt{x} + C$

(2) $\int \left( \frac{1}{x} + \frac{1}{x^2} \right) dx = \int \frac{1}{x} dx + \int x^{-2} dx = \ln |x| + \frac{x^{-1}}{-1} + C = \ln |x| - \frac{1}{x} + C$

(3) $\int \sqrt{x} dx = \int x^{\frac{1}{2}} dx = \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + C = \frac{2}{3} x^{\frac{3}{2}} + C = \frac{2}{3} x\sqrt{x} + C$

(4) $\int \frac{1}{x^2 + 3} dx = \int \frac{1}{x^2 + (\sqrt{3})^2} dx = \frac{1}{\sqrt{3}} \arctan \left( \frac{x}{\sqrt{3}} \right) + C$

$\int \frac{1}{x^2+a^2}dx = \frac{1}{a}\arctan(\frac{x}{a})+C$

3. 最終的な答え

(1) $2\sqrt{x} + C$

(2) $\ln |x| - \frac{1}{x} + C$

(3) $\frac{2}{3} x\sqrt{x} + C$

(4) $\frac{1}{\sqrt{3}} \arctan \left( \frac{x}{\sqrt{3}} \right) + C$