与えられた9つの積分を計算します。 (1) $\int \frac{x^{10}}{x^3} dx$ (2) $\int \sqrt[3]{x^4} dx$ (3) $\int \frac{x}{\sqrt[3]{x}} dx$ (4) $\int \frac{1}{x^2\sqrt[5]{x}} dx$ (5) $\int (x-4)^6 dx$ (6) $\int \frac{1}{\sqrt[3]{x+5}} dx$ (7) $\int 7x\sqrt{x^3} dx$ (8) $\int \frac{1}{2x\sqrt[3]{x}} dx$ (9) $\int \frac{7x}{\sqrt{x}\sqrt[3]{x}} dx$

1. 問題の内容

与えられた9つの積分を計算します。

(1) $\int \frac{x^{10}}{x^3} dx$

(2) $\int \sqrt[3]{x^4} dx$

(3) $\int \frac{x}{\sqrt[3]{x}} dx$

(4) $\int \frac{1}{x^2\sqrt[5]{x}} dx$

(5) $\int (x-4)^6 dx$

(6) $\int \frac{1}{\sqrt[3]{x+5}} dx$

(7) $\int 7x\sqrt{x^3} dx$

(8) $\int \frac{1}{2x\sqrt[3]{x}} dx$

(9) $\int \frac{7x}{\sqrt{x}\sqrt[3]{x}} dx$

2. 解き方の手順

(1) $\int \frac{x^{10}}{x^3} dx = \int x^7 dx = \frac{x^8}{8} + C$

(2) $\int \sqrt[3]{x^4} dx = \int x^{\frac{4}{3}} dx = \frac{x^{\frac{7}{3}}}{\frac{7}{3}} + C = \frac{3}{7}x^{\frac{7}{3}} + C$

(3) $\int \frac{x}{\sqrt[3]{x}} dx = \int \frac{x}{x^{\frac{1}{3}}} dx = \int x^{\frac{2}{3}} dx = \frac{x^{\frac{5}{3}}}{\frac{5}{3}} + C = \frac{3}{5}x^{\frac{5}{3}} + C$

(4) $\int \frac{1}{x^2\sqrt[5]{x}} dx = \int \frac{1}{x^2 x^{\frac{1}{5}}} dx = \int \frac{1}{x^{\frac{11}{5}}} dx = \int x^{-\frac{11}{5}} dx = \frac{x^{-\frac{6}{5}}}{-\frac{6}{5}} + C = -\frac{5}{6}x^{-\frac{6}{5}} + C$

(5) $\int (x-4)^6 dx = \frac{(x-4)^7}{7} + C$

(6) $\int \frac{1}{\sqrt[3]{x+5}} dx = \int (x+5)^{-\frac{1}{3}} dx = \frac{(x+5)^{\frac{2}{3}}}{\frac{2}{3}} + C = \frac{3}{2}(x+5)^{\frac{2}{3}} + C$

(7) $\int 7x\sqrt{x^3} dx = \int 7x x^{\frac{3}{2}} dx = \int 7x^{\frac{5}{2}} dx = 7 \frac{x^{\frac{7}{2}}}{\frac{7}{2}} + C = 2x^{\frac{7}{2}} + C$

(8) $\int \frac{1}{2x\sqrt[3]{x}} dx = \int \frac{1}{2x x^{\frac{1}{3}}} dx = \int \frac{1}{2x^{\frac{4}{3}}} dx = \frac{1}{2} \int x^{-\frac{4}{3}} dx = \frac{1}{2} \frac{x^{-\frac{1}{3}}}{-\frac{1}{3}} + C = -\frac{3}{2}x^{-\frac{1}{3}} + C$

(9) $\int \frac{7x}{\sqrt{x}\sqrt[3]{x}} dx = \int \frac{7x}{x^{\frac{1}{2}}x^{\frac{1}{3}}} dx = \int \frac{7x}{x^{\frac{5}{6}}} dx = \int 7x^{\frac{1}{6}} dx = 7 \frac{x^{\frac{7}{6}}}{\frac{7}{6}} + C = 6x^{\frac{7}{6}} + C$

3. 最終的な答え

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

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

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

(4) $-\frac{5}{6}x^{-\frac{6}{5}} + C$

(5) $\frac{(x-4)^7}{7} + C$

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

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

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

(9) $6x^{\frac{7}{6}} + C$