以下の定積分を計算します。 (1) $\int_{1}^{4} \frac{1}{\sqrt{x}} dx$ (2) $\int_{0}^{\pi} \cos t dt$ (3) $\int_{-1}^{1} u du$ (4) $\int_{0}^{1} \sqrt[3]{x^2} dx$ (5) $\int_{0}^{\frac{\pi}{4}} \frac{1}{\cos^2 x} dx$ (6) $\int_{-1}^{0} \frac{1}{e^x} dx$ (7) $\int_{0}^{1} e^{x-1} dx$ (8) $\int_{1}^{2} 2^x dx$ (9) $\int_{0}^{\frac{\pi}{2}} \sin^2 x dx$

はい、承知いたしました。画像にある数学の問題を解きます。

1. 問題の内容

以下の定積分を計算します。

(1) $\int_{1}^{4} \frac{1}{\sqrt{x}} dx$

(2) $\int_{0}^{\pi} \cos t dt$

(3) $\int_{-1}^{1} u du$

(4) $\int_{0}^{1} \sqrt[3]{x^2} dx$

(5) $\int_{0}^{\frac{\pi}{4}} \frac{1}{\cos^2 x} dx$

(6) $\int_{-1}^{0} \frac{1}{e^x} dx$

(7) $\int_{0}^{1} e^{x-1} dx$

(8) $\int_{1}^{2} 2^x dx$

(9) $\int_{0}^{\frac{\pi}{2}} \sin^2 x dx$

2. 解き方の手順

(1) $\int_{1}^{4} \frac{1}{\sqrt{x}} dx = \int_{1}^{4} x^{-\frac{1}{2}} dx$

$= [2x^{\frac{1}{2}}]_{1}^{4} = 2\sqrt{4} - 2\sqrt{1} = 4 - 2 = 2$

(2) $\int_{0}^{\pi} \cos t dt = [\sin t]_{0}^{\pi} = \sin \pi - \sin 0 = 0 - 0 = 0$

(3) $\int_{-1}^{1} u du = [\frac{1}{2}u^2]_{-1}^{1} = \frac{1}{2}(1)^2 - \frac{1}{2}(-1)^2 = \frac{1}{2} - \frac{1}{2} = 0$

(4) $\int_{0}^{1} \sqrt[3]{x^2} dx = \int_{0}^{1} x^{\frac{2}{3}} dx = [\frac{3}{5}x^{\frac{5}{3}}]_{0}^{1} = \frac{3}{5}(1)^{\frac{5}{3}} - \frac{3}{5}(0)^{\frac{5}{3}} = \frac{3}{5} - 0 = \frac{3}{5}$

(5) $\int_{0}^{\frac{\pi}{4}} \frac{1}{\cos^2 x} dx = \int_{0}^{\frac{\pi}{4}} \sec^2 x dx = [\tan x]_{0}^{\frac{\pi}{4}} = \tan \frac{\pi}{4} - \tan 0 = 1 - 0 = 1$

(6) $\int_{-1}^{0} \frac{1}{e^x} dx = \int_{-1}^{0} e^{-x} dx = [-e^{-x}]_{-1}^{0} = -e^{0} - (-e^{-(-1)}) = -1 + e = e-1$

(7) $\int_{0}^{1} e^{x-1} dx = [e^{x-1}]_{0}^{1} = e^{1-1} - e^{0-1} = e^0 - e^{-1} = 1 - \frac{1}{e}$

(8) $\int_{1}^{2} 2^x dx = [\frac{2^x}{\ln 2}]_{1}^{2} = \frac{2^2}{\ln 2} - \frac{2^1}{\ln 2} = \frac{4}{\ln 2} - \frac{2}{\ln 2} = \frac{2}{\ln 2}$

(9) $\int_{0}^{\frac{\pi}{2}} \sin^2 x dx = \int_{0}^{\frac{\pi}{2}} \frac{1 - \cos(2x)}{2} dx = [\frac{x}{2} - \frac{\sin(2x)}{4}]_{0}^{\frac{\pi}{2}} = (\frac{\pi}{4} - \frac{\sin(\pi)}{4}) - (0 - \frac{\sin(0)}{4}) = \frac{\pi}{4} - 0 - 0 + 0 = \frac{\pi}{4}$

3. 最終的な答え

(1) 2

(2) 0

(3) 0

(4) $\frac{3}{5}$

(5) 1

(6) $e-1$

(7) $1 - \frac{1}{e}$

(8) $\frac{2}{\ln 2}$

(9) $\frac{\pi}{4}$