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Hesap Modülleri

### $\sqrt{ax+b}$ içeren integraller

1.
$$\int{\frac{dx}{\sqrt{ax+b}}}=\frac{2\sqrt{ax+b}}{a}$$

2.
$$\int{\frac{xdx}{\sqrt{ax+b}}}=\frac{2(ax-2b)}{3a^2}\sqrt{ax+b}$$

3.
$$\int{\frac{x^2dx}{\sqrt{ax+b}}}=\frac{2(3a^2x^2-4abx+8b^2)}{15a^3}\sqrt{ax+b}$$

4.
$$\int{\frac{dx}{x(ax+b)}}= \left\{\begin{matrix} \frac{1}{\sqrt{b}}\ln{\left( \frac{\sqrt{ax+b}-\sqrt{b}}{\sqrt{ax+b}+\sqrt{b}} \right)}\\ \\frac{2}{\sqrt{-b}}\tan^{-1}\sqrt{\frac{ax+b}{-b}} \end{matrix}\right.$$

5.
$$\int{\frac{dx}{x^2\sqrt{ax+b}}}= -\frac{\sqrt{ax+b}}{bx}-\frac{a}{2b}\int{\frac{dx}{x\sqrt{ax+b}}}$$

6.
$$\int{\sqrt{ax+b}dx}=\frac{2\sqrt{(ax+b)^3}}{3a}$$

7.
$$\int{x\sqrt{ax+b}dx}=\frac{2(3ax-2b)}{15a^2}\sqrt{(ax+b)^3}$$

8.
$$\int{x^2\sqrt{ax+b}dx}=\frac{2(15a^2x^2-12abx+8b^2)}{105a^3}\sqrt{(ax+b)^3}$$

9.
$$\int{\frac{\sqrt{ax+b}}{x}dx}=2\sqrt{ax+b}+b\int\frac{dx}{x\sqrt{ax+b}}$$

10.
$$\int{\frac{\sqrt{ax+b}}{x^2}dx}=-\frac{\sqrt{ax+b}}{x}+\frac{a}{2}\int{\frac{dx}{x\sqrt{ax+b}}}$$

11.
$$\int{\frac{x^m}{\sqrt{ax+b}}dx}=\frac{2x^m\sqrt{ax+b}}{(2m+1)a}-\frac{2mb}{(2m+1)a}\int{\frac{x^{m-1}}{\sqrt{ax+b}}dx}$$

12.
$$\int{\frac{dx}{x^m\sqrt{ax+b}}}=-\frac{\sqrt{ax+b}}{(m-1)bx^{m-1}}-\frac{(2m-3)a}{(2m-2)b}\int{\frac{dx}{x^{m-1}\sqrt{ax+b}}}$$

13.
$$\int{x^m\sqrt{ax+b}dx}=\frac{2x^m}{(2m+3)}(ax+b)^{3/2}-\frac{2mb}{(2m+3)a}\int{x^{m-1}\sqrt{ax+b}dx}$$

14.
$$\int{\frac{\sqrt{ax+b}}{x^m}}=-\frac{\sqrt{ax+b}}{(m-1)x^{m-1}}+\frac{a}{2(m-1)}\int{\frac{dx}{x^{m-1}\sqrt{ax+b}}}$$

15.
$$\int{(ax+b)^{m/2}}=\frac{2(ax+b)^{(m+2)/2}}{a(m+2)}$$

16.
$$\int{x(ax+b)^{m/2}}=\frac{2(ax+b)^{(m+4)/2}}{a^2(m+4)}-\frac{2b(ax+b)^{(m+2)/2}}{a^2(m+2)}$$

17.
$$\int{x^2(ax+b)^{m/2}}=\frac{2(ax+b)^{(m+6)/2}}{a^3(m+6)}-\frac{4b(ax+b)^{(m+4)/2}}{a^3(m+4)}+\frac{2b^2(ax+b)^{(m+2)/2}}{a^3(m+2)}$$

18.
$$\int{\frac{(ax+b)^{m/2}}{x}dx}=\frac{2(ax+b)^{m/2}}{m}+b\int{\frac{(ax+b)^{(m-2)/2}}{x}dx}$$

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