与えられた17個の不定積分を求める問題です。積分定数はCとします。

1. 問題の内容

与えられた17個の不定積分を求める問題です。積分定数はCとします。

2. 解き方の手順

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

$\int x^{-3} dx = \frac{x^{-2}}{-2} + C = -\frac{1}{2x^2} + C$

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

$\int x^{3/2} dx = \frac{x^{5/2}}{5/2} + C = \frac{2}{5}x^{5/2} + C$

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

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

(4) $\int (2^x - e^x) dx$

$\int 2^x dx - \int e^x dx = \frac{2^x}{\log 2} - e^x + C$

(5) $\int \tan x dx$

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

(6) $\int \tan^2 3x dx$

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

(7) $\int (2-3x)^3 dx$

$u = 2 - 3x$ と置くと、$du = -3 dx$。したがって、$dx = -\frac{1}{3}du$。

$\int u^3 (-\frac{1}{3}) du = -\frac{1}{3} \int u^3 du = -\frac{1}{3} \frac{u^4}{4} + C = -\frac{1}{12}(2-3x)^4 + C$

(8) $\int x\sqrt{1-x} dx$

$u = 1-x$ と置くと、$x = 1-u$。$du = -dx$ なので、$dx = -du$。

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

(9) $\int 8\sin^3 x \cos x dx$

$u = \sin x$ と置くと、$du = \cos x dx$。

$\int 8u^3 du = 8 \frac{u^4}{4} + C = 2u^4 + C = 2\sin^4 x + C$

(10) $\int x\log x dx$

部分積分法を使う。$u = \log x$, $dv = x dx$ とすると、$du = \frac{1}{x} dx$, $v = \frac{x^2}{2}$。

$\int x\log x dx = (\log x)(\frac{x^2}{2}) - \int (\frac{x^2}{2})(\frac{1}{x}) dx = \frac{x^2}{2}\log x - \int \frac{x}{2} dx = \frac{x^2}{2}\log x - \frac{x^2}{4} + C$

(11) $\int \log(2x-1) dx$

部分積分法を使う。$u = \log(2x-1)$, $dv = dx$ とすると、$du = \frac{2}{2x-1} dx$, $v = x$。

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

(12) $\int x^2 e^{-x} dx$

部分積分法を2回使う。

$\int x^2 e^{-x} dx = -x^2e^{-x} - \int (-2x)e^{-x} dx = -x^2e^{-x} + 2\int xe^{-x} dx = -x^2e^{-x} + 2(-xe^{-x} - \int (-e^{-x})dx) = -x^2e^{-x} - 2xe^{-x} + 2\int e^{-x} dx = -x^2e^{-x} - 2xe^{-x} - 2e^{-x} + C = -e^{-x}(x^2 + 2x + 2) + C$

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

$\int \frac{x+3}{(x+1)(x+2)} dx = \int (\frac{A}{x+1} + \frac{B}{x+2}) dx$

$x+3 = A(x+2) + B(x+1)$。

$x = -1$ のとき、$2 = A$。$x = -2$ のとき、$1 = -B$, $B = -1$。

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

(14) $\int \cos^2 x dx$

$\int \frac{1 + \cos 2x}{2} dx = \frac{1}{2} \int (1 + \cos 2x) dx = \frac{1}{2} (x + \frac{1}{2}\sin 2x) + C = \frac{1}{2}x + \frac{1}{4}\sin 2x + C$

(15) $\int \frac{1-\sin x}{\cos x} dx$

$\int (\frac{1}{\cos x} - \frac{\sin x}{\cos x}) dx = \int (\sec x - \tan x) dx = \log|\sec x + \tan x| - (-\log|\cos x|) + C = \log|\sec x + \tan x| + \log|\cos x| + C$

(16) $\int e^{\sqrt{x}} dx$

$t = \sqrt{x}$ と置くと、$x = t^2$。$dx = 2t dt$

$\int e^t 2t dt = 2\int te^t dt$ (部分積分)

$u = t$, $dv = e^t dt$ とすると、$du = dt$, $v = e^t$

$2(te^t - \int e^t dt) = 2(te^t - e^t) + C = 2e^{\sqrt{x}}(\sqrt{x} - 1) + C$

(17) $\int \log(x + \sqrt{x^2+1}) dx$

$u = \log(x + \sqrt{x^2+1})$ と置くと、$du = \frac{1 + \frac{2x}{2\sqrt{x^2+1}}}{x+\sqrt{x^2+1}}dx=\frac{\frac{\sqrt{x^2+1}+x}{\sqrt{x^2+1}}}{x+\sqrt{x^2+1}}dx=\frac{1}{\sqrt{x^2+1}}dx$

$\int \log(x + \sqrt{x^2+1}) dx$の部分積分法で考える．$u=\log(x + \sqrt{x^2+1})$, $dv=dx$とすると，$du = \frac{1}{\sqrt{x^2+1}}dx$, $v = x$となる．

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

3. 最終的な答え

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

(2) $\frac{2}{5}x^{5/2} + C$

(3) $\sin x - \tan x + C$

(4) $\frac{2^x}{\log 2} - e^x + C$

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

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

(7) $-\frac{1}{12}(2-3x)^4 + C$

(8) $\frac{2}{3}(1-x)^{3/2} - \frac{2}{5}(1-x)^{5/2} + C$

(9) $2\sin^4 x + C$

(10) $\frac{x^2}{2}\log x - \frac{x^2}{4} + C$

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

(12) $-e^{-x}(x^2 + 2x + 2) + C$

(13) $2\log|x+1| - \log|x+2| + C$

(14) $\frac{1}{2}x + \frac{1}{4}\sin 2x + C$

(15) $\log|\sec x + \tan x| + \log|\cos x| + C$

(16) $2e^{\sqrt{x}}(\sqrt{x} - 1) + C$

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