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Hesap Modülleri

### $ax+b$ içeren integraller

1.
$$\int{\frac{dx}{ax+b}}=\frac{1}{a}\ln(ax+b)$$

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

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

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

5.
$$\int{\frac{dx}{x(ax+b)}}=\frac{1}{b}\ln(\frac{x}{ax+b})$$

6.
$$\int{\frac{dx}{x^2(ax+b)}}=-\frac{1}{bx}+\frac{a}{b^2}\ln\left(\frac{ax+b}{x}\right)$$

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

8.
$$\int{\frac{dx}{(ax+b)^2}}=\frac{-1}{a(ax+b)}$$

9.
$$\int{\frac{xdx}{(ax+b)^2}}=\frac{b}{a^2(ax+b)}+\frac{1}{a^2}\ln\left(ax+b\right)$$

10.
$$\int{\frac{x^2dx}{(ax+b)^2}}=\frac{ax+b}{a^3}-\frac{b^2}{a^3(ax+b)}-\frac{2b}{a^3}\ln\left(ax+b\right)$$

11.
$$\int{\frac{x^3dx}{(ax+b)^2}}=\frac{(ax+b)^2}{2a^4}-\frac{3b(ax+b)}{a^4}+\frac{b^3}{a^4(ax+b)}+\frac{3b^2}{a^4}\ln\left(ax+b\right)$$

12.
$$\int{\frac{dx}{x(ax+b)^2}}=\frac{1}{b(ax+b)}+\frac{1}{b^2}\ln\left(\frac{x}{ax+b}\right)$$

13.
$$\int{\frac{dx}{x^2(ax+b)^2}}=\frac{-a}{b^2(ax+b)}-\frac{1}{b^2x}+\frac{2a}{b^3}\ln\left(\frac{ax+b}{x}\right)$$

14.
$$\int{\frac{dx}{x^3(ax+b)^2}}=-\frac{(ax+b)^2}{2b^4x^2}+\frac{3a(ax+b)}{b^4x}-\frac{a^3x}{b^4(ax+b)}-\frac{3a^2}{b^4}\ln\left(\frac{ax+b}{x}\right)$$

15.
$$\int{\frac{dx}{(ax+b)^3}}=\frac{-1}{2a(ax+b)^2}$$

16.
$$\int{\frac{xdx}{(ax+b)^3}}=\frac{-1}{a^2(ax+b)}+\frac{b}{2a^2(ax+b)^2}$$

17.
$$\int{\frac{x^2dx}{(ax+b)^3}}=\frac{2b}{a^3(ax+b)}-\frac{b^2}{2a^3(ax+b)^2}+\frac{1}{a^3}\ln(ax+b)$$

18.
$$\int{\frac{x^3dx}{(ax+b)^3}}=\frac{x}{a^3}-\frac{3b^2}{a^4(ax+b)}+\frac{b^3}{2a^4(ax+b)^2}-\frac{3b}{a^4}\ln(ax+b)$$

19.
$$\int{\frac{dx}{x(ax+b)^3}}=\frac{a^2x^2}{2b^3(ax+b)^2}-\frac{2ax}{b^3(ax+b)}-\frac{1}{b^3}\ln\left(\frac{ax+b}{x}\right)$$

20.
$$\int{\frac{dx}{x^2(ax+b)^3}}=\frac{-a}{2b^2(ax+b)^2}-\frac{2a}{b^3(ax+b)}-\frac{1}{b^3x}+\frac{3a}{b^4}\ln\left(\frac{ax+b}{x}\right)$$

21.
$$\int{\frac{dx}{x^3(ax+b)^3}}=\frac{a^4x^2}{2b^5(ax+b)^2}-\frac{4a^3x}{b^5(ax+b)}-\frac{(ax+b)^2}{2b^5x^2}-\frac{6a^2}{b^5}\ln\left(\frac{ax+b}{x}\right)$$

22.
$$\int{(ax+b)^ndx}=\frac{(ax+b)^{n+1}}{(n+1)a},\quad n\neq-1$$

23.
$$\int{x(ax+b)^ndx}=\frac{(ax+b)^{n+2}}{(n+2)a^2}-\frac{b(ax+b)^{n+1}}{(n+1)a^2},\quad n\neq-1,-2$$

24.
$$\int{x^2(ax+b)^ndx}=\frac{(ax+b)^{n+3}}{(n+3)a^3}-\frac{2b(ax+b)^{n+2}}{(n+2)a^3}+\frac{b^2(ax+b)^{n+1}}{(n+1)a^3},$$

25.
$$\int x^{m}(ax+b)^{n}\,dx=\left\{\begin{matrix} \frac{x^{m+1}\left ( ax+b \right )^{n}}{m+n+1}+\frac{nb}{m+n+1}\int x^m\left ( ax+b \right )^{n-1}dx\\ \frac{x^m\left ( ax+b \right )^{n+1}}{\left ( m+n+1 \right )a}-\frac{mb}{\left ( m+n+1 \right )a}\int x^{m-1}\left ( ax+b \right )^ndx\\\frac{-x^{m+1}\left ( ax+b \right )^{n+1}}{\left ( n+1 \right )^b}+\frac{m+n+2}{\left ( n+1 \right )b}\int x^m\left ( ax+b \right )^{n+1} dx \end{matrix}\right.$$

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