次の不定積分を求めよ。 (1) $\int (x-1)^{2/3} dx$ (2) $\int (2x+1)^{-1/4} dx$ (3) $\int \sinh x dx$ (4) $\int \frac{dx}{3x+2}$ (5) $\int \frac{dx}{\sqrt{2-x^2}}$ (6) $\int \frac{dx}{2+x^2}$

1. 問題の内容

次の不定積分を求めよ。

(1) $\int (x-1)^{2/3} dx$

(2) $\int (2x+1)^{-1/4} dx$

(3) $\int \sinh x dx$

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

(5) $\int \frac{dx}{\sqrt{2-x^2}}$

(6) $\int \frac{dx}{2+x^2}$

2. 解き方の手順

(1) $\int (x-1)^{2/3} dx$

$u = x-1$とおくと、$du = dx$。

$\int u^{2/3} du = \frac{u^{5/3}}{5/3} + C = \frac{3}{5} u^{5/3} + C = \frac{3}{5} (x-1)^{5/3} + C$

(2) $\int (2x+1)^{-1/4} dx$

$u = 2x+1$とおくと、$du = 2dx$、$dx = \frac{1}{2}du$。

$\int u^{-1/4} \frac{1}{2} du = \frac{1}{2} \int u^{-1/4} du = \frac{1}{2} \frac{u^{3/4}}{3/4} + C = \frac{1}{2} \cdot \frac{4}{3} u^{3/4} + C = \frac{2}{3} u^{3/4} + C = \frac{2}{3} (2x+1)^{3/4} + C$

(3) $\int \sinh x dx$

$\int \sinh x dx = \cosh x + C$

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

$u = 3x+2$とおくと、$du = 3dx$、$dx = \frac{1}{3}du$。

$\int \frac{1}{u} \frac{1}{3} du = \frac{1}{3} \int \frac{1}{u} du = \frac{1}{3} \ln |u| + C = \frac{1}{3} \ln |3x+2| + C$

(5) $\int \frac{dx}{\sqrt{2-x^2}}$

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

$u = \frac{x}{\sqrt{2}}$とおくと、$x = \sqrt{2} u$、$dx = \sqrt{2} du$。

$\int \frac{\sqrt{2} du}{\sqrt{2-2u^2}} = \int \frac{\sqrt{2} du}{\sqrt{2}\sqrt{1-u^2}} = \int \frac{du}{\sqrt{1-u^2}} = \arcsin u + C = \arcsin \frac{x}{\sqrt{2}} + C$

(6) $\int \frac{dx}{2+x^2}$

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

$u = \frac{x}{\sqrt{2}}$とおくと、$x = \sqrt{2}u$、$dx = \sqrt{2} du$。

$\frac{1}{2} \int \frac{\sqrt{2} du}{1+u^2} = \frac{\sqrt{2}}{2} \int \frac{du}{1+u^2} = \frac{\sqrt{2}}{2} \arctan u + C = \frac{\sqrt{2}}{2} \arctan \frac{x}{\sqrt{2}} + C = \frac{1}{\sqrt{2}} \arctan \frac{x}{\sqrt{2}} + C$

3. 最終的な答え

(1) $\frac{3}{5} (x-1)^{5/3} + C$

(2) $\frac{2}{3} (2x+1)^{3/4} + C$

(3) $\cosh x + C$

(4) $\frac{1}{3} \ln |3x+2| + C$

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

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