Common Integrals INTEGRALS CONTAINING ax+b

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Common Integrals INTEGRALS CONTAINING ax+b

1.
2.: $displaystyle intdisplaystyle rac{x,dx}{ax+b}=displaystyle rac{x}{a}-displaystyle rac{b}{a^{2}}ln(ax+b)$
3.: $displaystyle intdisplaystyle rac{x^3,dx}{ax+b}=displaystyle rac{(ax+b)... ...splaystyle rac{3b^{2}(ax+b)}{a^{4}}- displaystyle rac{b^{3}}{a^4}ln(ax+b)$
4.: $displaystyle intdisplaystyle rac{x^2,dx}{ax+b}=displaystyle rac{(ax+b)... ...displaystyle rac{2b(ax+b)}{a^{3}}+displaystyle rac{b^{2}}{a^{3}}ln(ax+b)$
5.
6.: $displaystyle intdisplaystyle rac{dx}{x^{2}(ax+b)}=-displaystyle rac{1}{bx}+displaystyle rac{a}{b^{2}}lnleft(displaystyle rac{ax+b}{x} ight)$
7.: $displaystyle intdisplaystyle rac{dx}{x^{3}(ax+b)}=displaystyle rac{2ax-... ...}+displaystyle rac{a^{2}}{b^{3}}lnleft(displaystyle rac{x}{ax+b} ight)$
8.: $displaystyle intdisplaystyle rac{dx}{(ax+b)^{2}}=displaystyle rac{-1}{a(ax+b)}$
9.: $displaystyle intdisplaystyle rac{x,dx}{(ax+b)^{2}}=displaystyle rac{b}{a^{2}(ax+b)}+displaystyle rac{1}{a^{2}}ln(ax+b)$
10.: $displaystyle intdisplaystyle rac{x^{2},dx}{(ax+b)^{2}}=displaystyle ra... ...displaystyle rac{b^{2}}{a^{3}(ax+b)}-displaystyle rac{2b}{a^{3}}ln(ax+b)$
11.: $displaystyle intdisplaystyle rac{x^{3},dx}{(ax+b)^{2}}=displaystyle ra... ...playstyle rac{b^{3}}{a^{4}(ax+b)}+displaystyle rac{3b^{2}}{a^{4}}ln(ax+b)$
12.: $displaystyle intdisplaystyle rac{dx}{x(ax+b)}=displaystyle rac{1}{b(ax+b)}+displaystyle rac{1}{b^{2}}lnleft(displaystyle rac{x}{ax+b} ight)$
13.: $displaystyle intdisplaystyle rac{dx}{x^{2}(ax+b)^{2}}=displaystyle rac{... ...2}x}+displaystyle rac{2a}{b^{3}}lnleft(displaystyle rac{ax+b}{x} ight)$
14.: $displaystyle intdisplaystyle rac{dx}{x^{3}(ax+b)^{2}}=-displaystyle rac... ...-displaystyle rac{3a^{2}}{b^{4}}lnleft(displaystyle rac{ax+b}{x} ight)$
15.: $displaystyle intdisplaystyle rac{dx}{(ax+b)^{3}}=displaystyle rac{-1}{2(ax+b)^{2}}$
16.: $displaystyle intdisplaystyle rac{x,dx}{(ax+b)^{3}}=displaystyle rac{-1}{a^{2}(ax+b)}+displaystyle rac{b}{2a^{2}(ax+b)^{2}}$
17.: $displaystyle intdisplaystyle rac{x^{2},dx}{(ax+b)^{3}}=displaystyle ra... ...playstyle rac{b^{2}}{2a^{3}(ax+b)^{2}}+displaystyle rac{1}{a^{3}}ln(ax+b)$
18.: $displaystyle intdisplaystyle rac{x^{3},dx}{(ax+b)^{3}}=displaystyle ra... ...splaystyle rac{b^3}{2a^{4}(ax+b)^{2}}-displaystyle rac{3b}{a^{4}}ln(ax+b)$
19.: $displaystyle intdisplaystyle rac{dx}{x(ax+b)^{3}}=displaystyle rac{a^{2... ...x+b)}-displaystyle rac{1}{b^{3}}lnleft(displaystyle rac{ax+b}{x} ight)$
20.: $displaystyle intdisplaystyle rac{dx}{x^{2}(ax+b)^{3}}=displaystyle rac{... ...3}x}+displaystyle rac{3a}{b^{4}}lnleft(displaystyle rac{ax+b}{x} ight)$
21.: $displaystyle intdisplaystyle rac{dx}{x^{3}(ax+b)^{3}}=displaystyle rac{... ...-displaystyle rac{6a^{2}}{b^{5}}lnleft(displaystyle rac{ax+b}{x} ight)$
22.: $displaystyle int(ax+b)^{displaystyle n},dx=displaystyle rac{(ax+b)^{displaystyle n+1}}{(n+1)a}, ,;; n eq -1$
23.: $displaystyle int x(ax+b)^{displaystyle n},dx = displaystyle rac{(ax+b)^{... ...e rac{b(ax+b)^{displaystyle n+1}}{(n+1)a^{displaystyle2}},;;;n eq -1,-2$
24.: $displaystyle int x^{displaystyle2}(ax+b)^{displaystyle n},dx=displaystyle... ...ystyle2}(ax+b)^{displaystyle n+1}}{(n+1)a^{displaystyle3}};,;n eq -1,-2,-3$
25.: $displaystyle int x^{displaystyle m}(ax+b)^{displaystyle n},dx=left{ eg... ... int x^{displaystyle m}(ax+b)^{displaystyle{n+1}},dx end{array} ight.$

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Common Integrals INTEGRALS CONTAINING ax+b