了解しました。問題集の関数積分を求めます。

了解しました。問題集の関数積分を求めます。

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1. 問題の内容**

与えられた関数を積分します。以下はそれぞれの関数です。

(1) $\frac{1}{\cos^2(3x+1)}$

(2) $\frac{1}{\sqrt{4-(2x+1)^2}}$

(3) $\frac{1}{x^2+4x+6}$

(4) $\tan x$

(5) $\frac{1+\cos x}{x+\sin x}$

(6) $x^2(x^3+1)^{\frac{5}{2}}$

(7) $\frac{x+1}{\sqrt{x^2+2x+3}}$

(8) $\frac{(\log x)^7}{x}$

(9) $\frac{x}{\sqrt{1-x^2}}$

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2. 解き方の手順**

(1) $\frac{1}{\cos^2(3x+1)} = \sec^2(3x+1)$

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

(2) $\frac{1}{\sqrt{4-(2x+1)^2}}$

$2x+1 = 2\sin\theta$ と置換する.

$2dx = 2\cos\theta d\theta \Rightarrow dx = \cos\theta d\theta$

$\int \frac{1}{\sqrt{4-4\sin^2\theta}}\cos\theta d\theta = \int \frac{1}{2\cos\theta}\cos\theta d\theta = \int \frac{1}{2} d\theta = \frac{1}{2}\theta + C = \frac{1}{2}\arcsin(\frac{2x+1}{2}) + C$

(3) $\frac{1}{x^2+4x+6} = \frac{1}{(x+2)^2+2}$

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

(4) $\tan x = \frac{\sin x}{\cos x}$

$\int \tan x dx = \int \frac{\sin x}{\cos x} dx = -\log|\cos x| + C$

(5) $\frac{1+\cos x}{x+\sin x}$

$\int \frac{1+\cos x}{x+\sin x} dx = \log|x+\sin x| + C$

(6) $x^2(x^3+1)^{\frac{5}{2}}$

$u = x^3+1$ と置換する.

$du = 3x^2 dx \Rightarrow x^2dx = \frac{1}{3}du$

$\int x^2(x^3+1)^{\frac{5}{2}} dx = \int \frac{1}{3}u^{\frac{5}{2}} du = \frac{1}{3} \cdot \frac{2}{7} u^{\frac{7}{2}} + C = \frac{2}{21}(x^3+1)^{\frac{7}{2}} + C$

(7) $\frac{x+1}{\sqrt{x^2+2x+3}}$

$u = x^2+2x+3$ と置換する.

$du = (2x+2) dx \Rightarrow (x+1)dx = \frac{1}{2}du$

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

(8) $\frac{(\log x)^7}{x}$

$u = \log x$ と置換する.

$du = \frac{1}{x} dx$

$\int \frac{(\log x)^7}{x} dx = \int u^7 du = \frac{1}{8}u^8 + C = \frac{1}{8}(\log x)^8 + C$

(9) $\frac{x}{\sqrt{1-x^2}}$

$u = 1-x^2$ と置換する.

$du = -2x dx \Rightarrow xdx = -\frac{1}{2}du$

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

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3. 最終的な答え**

(1) $\frac{1}{3} \tan(3x+1) + C$

(2) $\frac{1}{2}\arcsin(\frac{2x+1}{2}) + C$

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

(4) $-\log|\cos x| + C$

(5) $\log|x+\sin x| + C$

(6) $\frac{2}{21}(x^3+1)^{\frac{7}{2}} + C$

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

(8) $\frac{1}{8}(\log x)^8 + C$

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