積分の公式を使った問題(不定積分)

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1. 次の不定積分を解きなさい．

   • (*)

     ${\displaystyle {\displaystyle \int 4xdx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \left(3x^2+2x\right)}dx}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{4}{\sqrt{x}}}dx}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \left(3e^x+4^x\right)dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \left(2\sin x+\frac{1}{3}\cos x\right)dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int {\left(x+1\right)}^{5}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{1}{\sqrt{9-x^2}}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{1}{5x+2}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \left(2x+1\right)\left(x-3\right)dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{3x^2-10x-8}{x-4}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{1}{\sqrt{x-1}-\sqrt{x+1}}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{2x+2}{x^2+2x-4}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{1}{x^2-1}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{{\cos }^{-1}x}{\sqrt{1-x^2}}dx}}$

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   • (*)

     ${\displaystyle {\displaystyle \int 5^x}dx}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int e^{-2x}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{x^3+x+1}{x^2+1}}dx}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \sqrt{\frac{1}{3}x}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \left(3x-5\right)dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \sin 2xdx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{1}{\sqrt{x-6}}dx}}$

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   • (*)

     ${\displaystyle {\displaystyle \int {\sin }^3}xdx}$

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   • (*)

     ${\displaystyle {\displaystyle \int \tan xdx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{1}{{\left(2-x\right)}^4}dx}}$

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   • (*)

     ${\displaystyle {\displaystyle \int {\cos }^{2}xdx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int {\cos }^{3}x\sin xdx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \left(12x^5-15x^4\right)dx}}$

     ⇒解答

   • (*)

     $$ {\displaystyle {\displaystyle \int \left(5x-3\right)}\left(2x+2\right)dx}$$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int 2\sqrt{x-3}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \left(e^x-2+3x\right)dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \sin \left(x-\frac{\pi }{4}\right)dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \left(a{x}^3-b{x}^2+c\right)dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int {\left(x+3\right)}^{2}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{3}{2x-1}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \left(\frac{2}{x^2}+\frac{1}{x}\right)dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{x+4}{2x^2+11x+12}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \sqrt{3x+4}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int x{\left(x^2+1\right)}^{3}dx}}$

     ⇒解答

   • (*)

     $$ {\displaystyle {\displaystyle \int \frac{4x^5-2x^4+x^3}{x}}dx}$$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{x}{2\sqrt{x}}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \left(a\sin x-x^2\right)dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{1}{x^2+6x+9}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{1}{\sqrt{4-8x}}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int 3x{\left(x+4\right)}^{3}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{3x^2}{x-a}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int 2x\log xdx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int x\sqrt{3x^2-1}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{x^2-81}{x+9}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{2}{1-x^2}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle \int \frac{3}{\sqrt{x+1}-\sqrt{x}}dx}$

     ⇒解答

   • (*)

     ${\displaystyle \int \frac{3x^3+12x+1}{x^2+4}dx}$

     ⇒解答

   • (*)

     ${\displaystyle \int \frac{1}{x^2-4}dx}$

     ⇒解答

   • (*)

     ${\displaystyle \int \frac{1}{\sqrt{4-9x^2}}dx}$

     ⇒解答

   • (*)

     ${\displaystyle \int \frac{1}{{\left(3x+1\right)}^2+4}dx}$

     ⇒解答

   • (*)

     ${\displaystyle \int \frac{1}{{\cos }^{2}2x}dx}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int {\tan }^{2}2x}dx}$

     ⇒解答

   • (*)

     ${\displaystyle \int sin3xcos5xdx}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \sin 3x\sin 5x}dx}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \cos 3x\cos 5x}dx}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int {\sin }^{2}2x}dx}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int {\cos }^{2}2x}dx}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{1}{{\tan }^{2}2x}}dx}$

     ⇒解答

   • (*)

     ${\displaystyle \int {\left(3x^2-5x+2\right)}^2\left(6x-5\right)dx}$

     ⇒解答

   • (*)

     ${\displaystyle \int x\sqrt{3x-5}dx}$

     ⇒解答

   • (*)

     ${\displaystyle \int \frac{1}{3x-1}dx}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{6x-5}{3x^2-5x+2}}dx}$

     ⇒解答

   • (*)

     ${\displaystyle \int e^{3x}dx}$

     ⇒解答

   • (*)

     ${\displaystyle \int \frac{e^{2x}}{e^{2x}-1}dx}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{\log 2x}{x}}dx}$

     ⇒解答

   • (*)

     ${\displaystyle \int sin\left(5x+\frac{\pi }{3}\right)dx}$

     ⇒解答

   • (*)

     ${\displaystyle \int cos\left(5x+\frac{\pi }{3}\right)dx}$

     ⇒解答

   • (*)

     ${\displaystyle \int tan2xdx}$

     ⇒解答

   • (*)

     ${\displaystyle \int {sec}^2\left(5x+\frac{\pi }{3}\right)dx}$

     ⇒解答

   • (*)

     ${\displaystyle \int {\text{csc}}^2\left(5x+\frac{\mathrm{\pi }}{3}\right)dx}$

     ⇒解答

   • (*)

     ${\displaystyle \int \frac{x}{2x+1}dx}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \sin 2x\sin xdx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int x\sqrt{x+1}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{1}{\sqrt{x+2}-\sqrt{x-2}}dx}}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int {\tan }^{3}x{\sec }^{2}x}dx}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{{\sec }^{2}x}{{\tan }^{3}x}}dx}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int e^{\cos x}\sin x}dx}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \left(5x^4-3x^2+\frac{3\sqrt{x}}{2}\right)}dx}$

     ⇒解答

   • (*)

     ${\displaystyle {\displaystyle \int \frac{1}{\sqrt{1-4x^2}}}dx}$

     ⇒解答

   • (*)

     ${\displaystyle \int \left(x^3+2x^2-\sqrt{x}\right)dx}$

     ⇒解答

   • (*)

     ${\displaystyle \int \frac{1}{\sqrt{x^2+5}}dx}$

     ⇒解答

2. 次の不定積分を部分積分法を用いて解きなさい．

   • (**)

     ${\displaystyle \int x^3{e}^{x}dx}$

     ⇒解答

   • (**)

     ${\displaystyle \int 2x{e}^{2x}dx}$

     ⇒解答

   • (**)

     ${\displaystyle \int xsinxdx}$

     ⇒解答

   • (**)

     ${\displaystyle \int xcosxdx}$

     ⇒解答

   • (**)

     ${\displaystyle \int e^{x}sinxdx}$

     ⇒解答

   • (**)

     ${\displaystyle \int \sqrt{x^2+5}dx}$

     ⇒解答

   • (**)

     ${\displaystyle \int log2xdx}$

     ⇒解答

   • (**)

     ${\displaystyle \int log\left(x+1\right)dx}$

     ⇒解答

   • (**)

     ${\displaystyle \int 2log\left(2x+1\right)dx}$

     ⇒解答

   • (**)

     ${\displaystyle \int {\left(log2x\right)}^{3}dx}$

     ⇒解答

   • (**)

     ${\displaystyle {\displaystyle \int {\tan }^{-1}x}dx}$

     ⇒解答

   • (**)

     ${\displaystyle {\displaystyle \int x\log 3xdx}}$

     ⇒解答

   • (**)

     ${\displaystyle {\displaystyle \int {(\log x)}^2}dx}$

     ⇒解答

   • (**)

     ${\displaystyle \int x{e}^x\mathit{dx}}$

     ⇒解答

   • (**)

     ${\displaystyle \int e^{2x}sinxdx}$

     ⇒解答

3. 次の不定積分を解きなさい．
   1. ${\displaystyle \int \frac{1}{cosx}dx}$ を ${\displaystyle tan\frac{x}{2}=t}$ で置換して解きなさい．　⇒解答
   2. ${\displaystyle \int \frac{1}{sinx}dx}$ を ${\displaystyle tan\frac{x}{2}=t}$ で置換して解きなさい．　⇒解答
   3. ${\displaystyle \int \frac{1}{sinx+cosx+1}dx}$ を ${\displaystyle tan\frac{x}{2}=t}$ で置換して解きなさい．　⇒解答
4. 次の不定積分 ${\displaystyle \int sinxcosxdx}$ をそれぞれの条件で解きなさい．
   1. ${\displaystyle 2}$ 倍角の公式を用いて解きなさい．　⇒解答
   2. ${\displaystyle cosx=t}$ で置換して解きなさい．　⇒解答
   3. ${\displaystyle sinx=t}$ で置換して解きなさい．　⇒解答
   4. ${\displaystyle {\displaystyle \int {\left(-\cos x\right)}^{\prime }\cos x}dx}$ と考え部分積分を用いて解きなさい．　⇒解答
   5. ${\displaystyle \int sinx{\left(sinx\right)}^{\prime }dx}$ と考え部分積分を用いて解きなさい．　⇒解答

 

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最終更新日： 2025年4月27日