✨Danh sách tích phân với hàm vô tỉ

Dưới đây là danh sách tích phân (nguyên hàm) với hàm vô tỉ. Đối với danh sách đầy đủ hàm tích phân, xem danh sách tích phân. Trong bài này, hằng số tích phân được lược bớt đi cho ngắn gọn.

Tích phân chứa r =

• $\backslash int\; r\backslash ,dx\; =\; \backslash frac\{1\}\{2\}\backslash left(x\; r\; +a^2\backslash ,\backslash ln\backslash left(x+r\backslash right)\backslash right)$
• $\backslash int\; r^3\backslash ,dx\; =\; \backslash frac\{1\}\{4\}xr^3+\backslash frac\{3\}\{8\}a^2xr+\backslash frac\{3\}\{8\}a^4\backslash ln\backslash left(x+r\backslash right)$
• $\backslash int\; r^5\backslash ,dx\; =\; \backslash frac\{1\}\{6\}xr^5+\backslash frac\{5\}\{24\}a^2xr^3+\backslash frac\{5\}\{16\}a^4xr+\backslash frac\{5\}\{16\}a^6\backslash ln\backslash left(x+r\backslash right)$
• $\backslash int\; x\; r\backslash ,dx\; =\; \backslash frac\{r^3\}\{3\}$
• $\backslash int\; x\; r^3\backslash ,dx\; =\; \backslash frac\{r^5\}\{5\}$
• $\backslash int\; x\; r^\{2n+1\}\backslash ,dx\; =\; \backslash frac\{r^\{2n+3\{2n+3\}$
• $\backslash int\; x^2\; r\backslash ,dx\; =\; \backslash frac\{xr^3\}\{4\}-\backslash frac\{a^2xr\}\{8\}-\backslash frac\{a^4\}\{8\}\backslash ln\backslash left(x+r\backslash right)$
• $\backslash int\; x^2\; r^3\backslash ,dx\; =\; \backslash frac\{xr^5\}\{6\}-\backslash frac\{a^2xr^3\}\{24\}-\backslash frac\{a^4xr\}\{16\}-\backslash frac\{a^6\}\{16\}\backslash ln\backslash left(x+r\backslash right)$
• $\backslash int\; x^3\; r\backslash ,dx\; =\; \backslash frac\{r^5\}\{5\}\; -\; \backslash frac\{a^2\; r^3\}\{3\}$
• $\backslash int\; x^3\; r^3\backslash ,dx\; =\; \backslash frac\{r^7\}\{7\}-\backslash frac\{a^2r^5\}\{5\}$
• $\backslash int\; x^3\; r^\{2n+1\}\backslash ,dx\; =\; \backslash frac\{r^\{2n+5\{2n+5\}\; -\; \backslash frac\{a^2\; r^\{2n+3\{2n+3\}$
• $\backslash int\; x^4\; r\backslash ,dx\; =\; \backslash frac\{x^3r^3\}\{6\}-\backslash frac\{a^2xr^3\}\{8\}+\backslash frac\{a^4xr\}\{16\}+\backslash frac\{a^6\}\{16\}\backslash ln\backslash left(x+r\backslash right)$
• $\backslash int\; x^4\; r^3\backslash ,dx\; =\; \backslash frac\{x^3r^5\}\{8\}-\backslash frac\{a^2xr^5\}\{16\}+\backslash frac\{a^4xr^3\}\{64\}+\backslash frac\{3a^6xr\}\{128\}+\backslash frac\{3a^8\}\{128\}\backslash ln\backslash left(x+r\backslash right)$
• $\backslash int\; x^5\; r\backslash ,dx\; =\; \backslash frac\{r^7\}\{7\}\; -\; \backslash frac\{2\; a^2\; r^5\}\{5\}\; +\; \backslash frac\{a^4\; r^3\}\{3\}$
• $\backslash int\; x^5\; r^3\backslash ,dx\; =\; \backslash frac\{r^9\}\{9\}\; -\; \backslash frac\{2\; a^2\; r^7\}\{7\}\; +\; \backslash frac\{a^4\; r^5\}\{5\}$
• $\backslash int\; x^5\; r^\{2n+1\}\backslash ,dx\; =\; \backslash frac\{r^\{2n+7\{2n+7\}\; -\; \backslash frac\{2a^2r^\{2n+5\{2n+5\}+\backslash frac\{a^4\; r^\{2n+3\{2n+3\}$
• $\backslash int\backslash frac\{r\backslash ,dx\}\{x\}\; =\; r-a\backslash ln\backslash left|\backslash frac\{a+r\}\{x\}\backslash right|\; =\; r\; -\; a\backslash ,\; \backslash operatorname\{arsinh\}\backslash frac\{a\}\{x\}$
• $\backslash int\backslash frac\{r^3\backslash ,dx\}\{x\}\; =\; \backslash frac\{r^3\}\{3\}+a^2r-a^3\backslash ln\backslash left|\backslash frac\{a+r\}\{x\}\backslash right|$
• $\backslash int\backslash frac\{r^5\backslash ,dx\}\{x\}\; =\; \backslash frac\{r^5\}\{5\}+\backslash frac\{a^2r^3\}\{3\}+a^4r-a^5\backslash ln\backslash left|\backslash frac\{a+r\}\{x\}\backslash right|$
• $\backslash int\backslash frac\{r^7\backslash ,dx\}\{x\}\; =\; \backslash frac\{r^7\}\{7\}+\backslash frac\{a^2r^5\}\{5\}+\backslash frac\{a^4r^3\}\{3\}+a^6r-a^7\backslash ln\backslash left|\backslash frac\{a+r\}\{x\}\backslash right|$
• $\backslash int\backslash frac\{dx\}\{r\}\; =\; \backslash operatorname\{arsinh\}\backslash frac\{x\}\{a\}\; =\; \backslash ln\backslash left(\; \backslash frac\{x+r\}\{a\}\; \backslash right)$
• $\backslash int\backslash frac\{dx\}\{r^3\}\; =\; \backslash frac\{x\}\{a^2r\}$
• $\backslash int\backslash frac\{x\backslash ,dx\}\{r\}\; =\; r$
• $\backslash int\backslash frac\{x\backslash ,dx\}\{r^3\}\; =\; -\backslash frac\{1\}\{r\}$
• $\backslash int\backslash frac\{x^2\backslash ,dx\}\{r\}\; =\; \backslash frac\{x\}\{2\}r-\backslash frac\{a^2\}\{2\}\backslash ,\backslash operatorname\{arsinh\}\backslash frac\{x\}\{a\}\; =\; \backslash frac\{x\}\{2\}r-\backslash frac\{a^2\}\{2\}\backslash ln\backslash left(\; \backslash frac\; \{x+r\}\{a\}\; \backslash right)$
• $\backslash int\backslash frac\{dx\}\{xr\}\; =\; -\backslash frac\{1\}\{a\}\backslash ,\backslash operatorname\{arsinh\}\backslash frac\{a\}\{x\}\; =\; -\backslash frac\{1\}\{a\}\backslash ln\backslash left|\backslash frac\{a+r\}\{x\}\backslash right|$

Tích phân chứa s =

Giả định x² > a² (đối với trường hợp x² < a², xem mục sau):

• $\backslash int\; s\backslash ,dx\; =\; \backslash frac\{1\}\{2\}\backslash left(xs-a^\{2\}\backslash ln\backslash left|x+s\backslash right|\backslash right)$
• $\backslash int\; xs\backslash ,dx\; =\; \backslash frac\{1\}\{3\}s^3$
• $\backslash int\backslash frac\{s\backslash ,dx\}\{x\}\; =\; s\; -\; |a|\backslash arccos\backslash left|\backslash frac\{a\}\{x\}\backslash right|$
• $\backslash int\backslash frac\{dx\}\{s\}\; =\; \backslash ln\backslash left|\backslash frac\{x+s\}\{a\}\backslash right|\backslash ,.$ Ở đây $\backslash ln\backslash left|\backslash frac\{x+s\}\{a\}\backslash right|\; =\backslash operatorname\{sgn\}(x)\backslash ,\backslash operatorname\{arcosh\}\backslash left|\backslash frac\{x\}\{a\}\backslash right|\; =\backslash frac\{1\}\{2\}\backslash ln\backslash left(\backslash frac\{x+s\}\{x-s\}\backslash right)\backslash ,,$ trong đó $\backslash operatorname\{arcosh\}\backslash left|\backslash frac\{x\}\{a\}\backslash right|$ lấy giá trị dương.
• $\backslash int\backslash frac\{dx\}\{xs\}\; =\; \backslash frac\{1\}\{a\}\backslash operatorname\{arcsec\}\backslash left|\backslash frac\{x\}\{a\}\backslash right|$
• $\backslash int\backslash frac\{x\backslash ,dx\}\{s\}\; =\; s$
• $\backslash int\backslash frac\{x\backslash ,dx\}\{s^3\}\; =\; -\backslash frac\{1\}\{s\}$
• $\backslash int\backslash frac\{x\backslash ,dx\}\{s^5\}\; =\; -\backslash frac\{1\}\{3s^3\}$
• $\backslash int\backslash frac\{x\backslash ,dx\}\{s^7\}\; =\; -\backslash frac\{1\}\{5s^5\}$
• $\backslash int\backslash frac\{x\backslash ,dx\}\{s^\{2n+1\; =\; -\backslash frac\{1\}\{(2n-1)s^\{2n-1$
• $\backslash int\backslash frac\{x^\{2m\}\backslash ,dx\}\{s^\{2n+1\; =\; -\backslash frac\{1\}\{2n-1\}\backslash frac\{x^\{2m-1\{s^\{2n-1+\backslash frac\{2m-1\}\{2n-1\}\backslash int\backslash frac\{x^\{2m-2\}\backslash ,dx\}\{s^\{2n-1$
• $\backslash int\backslash frac\{x^2\backslash ,dx\}\{s\}\; =\; \backslash frac\{xs\}\{2\}+\backslash frac\{a^2\}\{2\}\backslash ln\backslash left|\backslash frac\{x+s\}\{a\}\backslash right|$
• $\backslash int\backslash frac\{x^2\backslash ,dx\}\{s^3\}\; =\; -\backslash frac\{x\}\{s\}+\backslash ln\backslash left|\backslash frac\{x+s\}\{a\}\backslash right|$
• $\backslash int\backslash frac\{x^4\backslash ,dx\}\{s\}\; =\; \backslash frac\{x^3s\}\{4\}+\backslash frac\{3\}\{8\}a^2xs+\backslash frac\{3\}\{8\}a^4\backslash ln\backslash left|\backslash frac\{x+s\}\{a\}\backslash right|$
• $\backslash int\backslash frac\{x^4\backslash ,dx\}\{s^3\}\; =\; \backslash frac\{xs\}\{2\}-\backslash frac\{a^2x\}\{s\}+\backslash frac\{3\}\{2\}a^2\backslash ln\backslash left|\backslash frac\{x+s\}\{a\}\backslash right|$
• $\backslash int\backslash frac\{x^4\backslash ,dx\}\{s^5\}\; =\; -\backslash frac\{x\}\{s\}-\backslash frac\{1\}\{3\}\backslash frac\{x^3\}\{s^3\}+\backslash ln\backslash left|\backslash frac\{x+s\}\{a\}\backslash right|$
• $\backslash int\backslash frac\{x^\{2m\}\backslash ,dx\}\{s^\{2n+1\; =\; (-1)^\{n-m\}\backslash frac\{1\}\{a^\{2(n-m)\backslash sum\_\{i=0\}^\{n-m-1\}\backslash frac\{1\}\{2(m+i)+1\}\{n-m-1\; \backslash choose\; i\}\backslash frac\{x^\{2(m+i)+1\{s^\{2(m+i)+1\backslash qquad\backslash mbox\{(\}n>m\backslash ge0\backslash mbox\{)\}$
• $\backslash int\backslash frac\{dx\}\{s^3\}\; =\; -\backslash frac\{1\}\{a^2\}\backslash frac\{x\}\{s\}$
• $\backslash int\backslash frac\{dx\}\{s^5\}\; =\; \backslash frac\{1\}\{a^4\}\backslash left[\backslash frac\{x\}\{s\}-\backslash frac\{1\}\{3\}\backslash frac\{x^3\}\{s^3\}\backslash right]$
• $\backslash int\backslash frac\{dx\}\{s^7\}\; =-\backslash frac\{1\}\{a^6\}\backslash left[\backslash frac\{x\}\{s\}-\backslash frac\{2\}\{3\}\backslash frac\{x^3\}\{s^3\}+\backslash frac\{1\}\{5\}\backslash frac\{x^5\}\{s^5\}\backslash right]$
• $\backslash int\backslash frac\{dx\}\{s^9\}\; =\backslash frac\{1\}\{a^8\}\backslash left[\backslash frac\{x\}\{s\}-\backslash frac\{3\}\{3\}\backslash frac\{x^3\}\{s^3\}+\backslash frac\{3\}\{5\}\backslash frac\{x^5\}\{s^5\}-\backslash frac\{1\}\{7\}\backslash frac\{x^7\}\{s^7\}\backslash right]$
• $\backslash int\backslash frac\{x^2\backslash ,dx\}\{s^5\}\; =\; -\backslash frac\{1\}\{a^2\}\backslash frac\{x^3\}\{3s^3\}$
• $\backslash int\backslash frac\{x^2\backslash ,dx\}\{s^7\}\; =\; \backslash frac\{1\}\{a^4\}\backslash left[\backslash frac\{1\}\{3\}\backslash frac\{x^3\}\{s^3\}-\backslash frac\{1\}\{5\}\backslash frac\{x^5\}\{s^5\}\backslash right]$
• $\backslash int\backslash frac\{x^2\backslash ,dx\}\{s^9\}\; =\; -\backslash frac\{1\}\{a^6\}\backslash left[\backslash frac\{1\}\{3\}\backslash frac\{x^3\}\{s^3\}-\backslash frac\{2\}\{5\}\backslash frac\{x^5\}\{s^5\}+\backslash frac\{1\}\{7\}\backslash frac\{x^7\}\{s^7\}\backslash right]$

Tích phân chứa u =

• $\backslash int\; u\backslash ,dx\; =\; \backslash frac\{1\}\{2\}\backslash left(xu+a^2\backslash arcsin\backslash frac\{x\}\{a\}\backslash right)\; \backslash qquad\backslash mbox\{(\}|x|\backslash leq|a|\backslash mbox\{)\}$
• $\backslash int\; xu\backslash ,dx\; =\; -\backslash frac\{1\}\{3\}\; u^3\; \backslash qquad\backslash mbox\{(\}|x|\backslash leq|a|\backslash mbox\{)\}$
• $\backslash int\; x^2u\backslash ,dx\; =\; -\backslash frac\{x\}\{4\}\; u^3+\backslash frac\{a^2\}\{8\}(xu+a^2\backslash arcsin\backslash frac\{x\}\{a\})\; \backslash qquad\backslash mbox\{(\}|x|\backslash leq|a|\backslash mbox\{)\}$
• $\backslash int\backslash frac\{u\backslash ,dx\}\{x\}\; =\; u-a\backslash ln\backslash left|\backslash frac\{a+u\}\{x\}\backslash right|\; \backslash qquad\backslash mbox\{(\}|x|\backslash leq|a|\backslash mbox\{)\}$
• $\backslash int\backslash frac\{dx\}\{u\}\; =\; \backslash arcsin\backslash frac\{x\}\{a\}\; \backslash qquad\backslash mbox\{(\}|x|\backslash leq|a|\backslash mbox\{)\}$
• $\backslash int\backslash frac\{x^2\backslash ,dx\}\{u\}\; =\; \backslash frac\{1\}\{2\}\backslash left(-xu+a^2\backslash arcsin\backslash frac\{x\}\{a\}\backslash right)\; \backslash qquad\backslash mbox\{(\}|x|\backslash leq|a|\backslash mbox\{)\}$
• $\backslash int\; u\backslash ,dx\; =\; \backslash frac\{1\}\{2\}\backslash left(xu-\backslash sgn\; x\backslash ,\backslash operatorname\{arcosh\}\backslash left|\backslash frac\{x\}\{a\}\backslash right|\backslash right)\; \backslash qquad\backslash mbox\{(với\; \}|x|\backslash ge|a|\backslash mbox\{)\}$
• $\backslash int\; \backslash frac\{x\}\{u\}\backslash ,dx\; =\; -u\; \backslash qquad\backslash mbox\{(\}|x|\backslash leq|a|\backslash mbox\{)\}$

Tích phân chứa R =

Giả định rằng (ax² + bx + c) không thể chuyển về dạng (px + q)² với p và q là hằng số.

• $\backslash int\backslash frac\{dx\}\{R\}\; =\; \backslash frac\{1\}\{\backslash sqrt\{a\backslash ln\backslash left|2\backslash sqrt\{a\}R+2ax+b\backslash right|\; \backslash qquad\; \backslash mbox\{(với\; \}a>0\backslash mbox\{)\}$
• $\backslash int\backslash frac\{dx\}\{R\}\; =\; \backslash frac\{1\}\{\backslash sqrt\{a\backslash ,\backslash operatorname\{arsinh\}\backslash frac\{2ax+b\}\{\backslash sqrt\{4ac-b^2\; \backslash qquad\; \backslash mbox\{(với\; \}a>0\backslash mbox\{,\; \}4ac-b^2>0\backslash mbox\{)\}$
• $\backslash int\backslash frac\{dx\}\{R\}\; =\; \backslash frac\{1\}\{\backslash sqrt\{a\backslash ln|2ax+b|\; \backslash quad\; \backslash mbox\{(với\; \}a>0\backslash mbox\{,\; \}4ac-b^2=0\backslash mbox\{)\}$
• $\backslash int\backslash frac\{dx\}\{R^3\}\; =\; \backslash frac\{4ax+2b\}\{(4ac-b^2)R\}$
• $\backslash int\backslash frac\{dx\}\{R^5\}\; =\; \backslash frac\{4ax+2b\}\{3(4ac-b^2)R\}\backslash left(\backslash frac\{1\}\{R^2\}+\backslash frac\{8a\}\{4ac-b^2\}\backslash right)$
• $\backslash int\backslash frac\{dx\}\{R^\{2n+1\; =\; \backslash frac\{2\}\{(2n-1)(4ac-b^2)\}\backslash left(\backslash frac\{2ax+b\}\{R^\{2n-1+4a(n-1)\backslash int\backslash frac\{dx\}\{R^\{2n-1\backslash right)$
• $\backslash int\backslash frac\{x\}\{R\}\backslash ,dx\; =\; \backslash frac\{R\}\{a\}-\backslash frac\{b\}\{2a\}\backslash int\backslash frac\{dx\}\{R\}$
• $\backslash int\backslash frac\{x\}\{R^3\}\backslash ,dx\; =\; -\backslash frac\{2bx+4c\}\{(4ac-b^2)R\}$
• $\backslash int\backslash frac\{x\}\{R^\{2n+1\backslash ,dx\; =\; -\backslash frac\{1\}\{(2n-1)aR^\{2n-1-\backslash frac\{b\}\{2a\}\backslash int\backslash frac\{dx\}\{R^\{2n+1$
• $\backslash int\backslash frac\{dx\}\{xR\}\; =\; -\backslash frac\{1\}\{\backslash sqrt\{c\backslash ln\; \backslash left|\backslash frac\{2\backslash sqrt\{c\}R+bx+2c\}\{x\}\backslash right|,\; ~\; c\; >\; 0$
• $\backslash int\backslash frac\{dx\}\{xR\}\; =\; -\backslash frac\{2\}\{bx\}\backslash left(\backslash sqrt\{ax^2+bx\}\backslash right),\; ~\; c\; =\; 0$
• $\backslash int\backslash frac\{x^2\}\{R\}\backslash ,dx\; =\; \backslash frac\{2ax-3b\}\{4a^2\}R+\backslash frac\{3b^2-4ac\}\{8a^2\}\backslash int\backslash frac\{dx\}\{R\}$
• $\backslash int\; \backslash frac\{dx\}\{x^\{2\}\; R\}\; =\; -\backslash frac\{\; R\}\{cx\}-\backslash frac\{b\}\{2c\}\; \backslash int\; \backslash frac\{dx\}\{x\; R\}$
• $\backslash int\; R\backslash ,dx\; =\; \backslash frac\{2ax+b\}\{4a\}\; R\; +\; \backslash frac\{4ac-b^\{2\{8a\}\; \backslash int\; \backslash frac\{dx\}\{\; R\}$
• $\backslash int\; x\; R\backslash ,dx\; =\; \backslash frac\{R^3\}\{3a\}-\backslash frac\{b(2ax+b)\}\{8a^\{2\; R\; -\; \backslash frac\{b(4ac-b^\{2\})\}\{16a^\{2\; \backslash int\; \backslash frac\{dx\}\{\; R\}$
• $\backslash int\; x^\{2\}\; R\backslash ,dx\; =\; \backslash frac\{6ax-5b\}\{24a^\{2R^3+\backslash frac\{5b^\{2\}-4ac\}\{16a^\{2\; \backslash int\; R\backslash ,dx$
• $\backslash int\; \backslash frac\{\; R\}\{x\}\backslash ,dx\; =\; R\; +\; \backslash frac\{b\}\{2\}\; \backslash int\; \backslash frac\{dx\}\{\; R\}+c\; \backslash int\; \backslash frac\{dx\}\{x\; R\}$
• $\backslash int\; \backslash frac\{\; R\}\{x^\{2\backslash ,dx\; =\; -\backslash frac\{\; R\}\{x\}+a\; \backslash int\; \backslash frac\{dx\}\{R\}+\; \backslash frac\{b\}\{2\}\; \backslash int\; \backslash frac\{dx\}\{\; xR\}$
• $\backslash int\; \backslash frac\{x^\{2\}\backslash ,dx\}\{R^3\}\; =\; \backslash frac\{(2b^\{2\}-4ac)x+2bc\}\{a(4ac-b^\{2\})\; R\}+\; \backslash frac\{1\}\{a\}\; \backslash int\; \backslash frac\{dx\}\{\; R\}$

Tích phân chứa S =

• $\backslash int\; S\backslash ,dx\; =\; \backslash frac\{2\; S^\{3\{3\; a\}$
• $\backslash int\; \backslash frac\{dx\}\{S\}\; =\; \backslash frac\{2S\}\{a\}$
• $\backslash int\; \backslash frac\{x^\{n\{S\}\backslash ,dx\; =\; \backslash frac\{2\}\{a\; (2\; n\; +\; 1)\}\; \backslash left(\; x^\{n\}\; S\; -\; b\; n\; \backslash int\; \backslash frac\{x^\{n\; -\; 1\{S\}\backslash ,dx\backslash right)$
• $\backslash int\; x^\{n\}\; S\backslash ,dx\; =\; \backslash frac\{2\}\{a\; (2\; n\; +\; 3)\}\; \backslash left(x^\{n\}\; S^\{3\}\; -\; n\; b\; \backslash int\; x^\{n\; -\; 1\}\; S\backslash ,dx\backslash right)$
• $\backslash int\; \backslash frac\{1\}\{x^\{n\}\; S\}\backslash ,dx\; =\; -\backslash frac\{1\}\{b\; (n\; -\; 1)\}\; \backslash left(\; \backslash frac\{S\}\{x^\{n\; -\; 1\; +\; \backslash left(\; n\; -\; \backslash frac\{3\}\{2\}\backslash right)\; a\; \backslash int\; \backslash frac\{dx\}\{x^\{n\; -\; 1\}\; S\}\backslash right)$