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$\displaystyle \int \dfrac{1}{1-\sin^2 x}\,\mathrm{d}x$

$\displaystyle \int \dfrac{1}{1-\sin^2 x}\,\mathrm{d}x=\int \dfrac{1}{\cos^2 x}\,\mathrm{d}x=\bbox[#FFECB3,5px]{\tan x+k}$

$\displaystyle \int \dfrac{7x^4-3x^2-3x+5}{4x^2}\,\mathrm{d}x$

$\displaystyle \int \dfrac{7x^4-3x^2-3x+5}{4x^2}\,\mathrm{d}x= \int \left(\dfrac{7}{4}x^2-\dfrac{3}{4}-\dfrac{3}{4}\dfrac{1}{x}+\dfrac{5}{4}\dfrac{1}{x^2}\right)\,\mathrm{d}x= \bbox[#FFECB3,5px]{\dfrac{7}{12}x^3-\dfrac{3}{4}x-\dfrac{3}{4}\ln|x|-\dfrac{5}{4x}+k}$

$\displaystyle \int \dfrac{1+7x^2-5x^3}{2\sqrt{x}}\,\mathrm{d}x$

$\displaystyle \int \dfrac{1+7x^2-5x^3}{2\sqrt{x}}\,\mathrm{d}x= \int \left(\dfrac{1}{2}x^{-\frac{1}{2}}+\dfrac{7}{2}x^{\frac{3}{2}}-\dfrac{5}{2}x^{\frac{5}{2}}\right)\,\mathrm{d}x= \dfrac{1}{2}\left(\dfrac{1}{2}\right)^{-1}x^{\frac{1}{2}}+\dfrac{7}{2}\left(\dfrac{5}{2}\right)^{-1}x^{\frac{5}{2}}-\dfrac{5}{2}\left(\dfrac{7}{2}\right)^{-1}x^{\frac{7}{2}}+k=\\ \qquad\displaystyle \bbox[#FFECB3,5px]{x^{\frac{1}{2}}+\dfrac{7}{5}x^{\frac{5}{2}}-\dfrac{5}{7}x^{\frac{7}{2}}+k}$

$\displaystyle \int \dfrac{3x+4\sqrt[4]{x}}{5\sqrt[3]{x}}\,\mathrm{d}x$

$\displaystyle \int \dfrac{3x+4\sqrt[4]{x}}{5\sqrt[3]{x}}\,\mathrm{d}x= \int \dfrac{3x+4x^{\frac{1}{4}}}{5x^{\frac{1}{3}}}\,\mathrm{d}x= \int \left(\dfrac{3}{5}x^{\frac{2}{3}}+\dfrac{4}{5}x^{-\frac{1}{12}}\right)\,\mathrm{d}x= \dfrac{3}{5}\left(\dfrac{5}{3}\right)^{-1}x^{\frac{5}{3}}+\dfrac{4}{5}\left(\dfrac{11}{12}\right)^{-1}x^{\frac{11}{12}}+k=\\ \qquad\displaystyle \bbox[#FFECB3,5px]{\dfrac{9}{25}x^{\frac{5}{3}}+\dfrac{48}{55}x^{\frac{11}{12}}+k}$

$\displaystyle \int \dfrac{4\sqrt{x}-3\sqrt[5]{x^3}}{3\sqrt[3]{x^2}}\,\mathrm{d}x$

$\displaystyle \int \dfrac{4\sqrt{x}-3\sqrt[5]{x^3}}{3\sqrt[3]{x^2}}\,\mathrm{d}x= \int \dfrac{4x^{\frac{1}{2}}-3x^{\frac{3}{5}}}{3x^{\frac{2}{3}}}\,\mathrm{d}x= \int \left(\dfrac{4}{3}x^{-\frac{1}{6}}-x^{-\frac{1}{15}}\right)\,\mathrm{d}x= \dfrac{4}{3}\left(\dfrac{5}{6}\right)^{-1}x^{\frac{5}{6}}-\left(\dfrac{14}{15}\right)^{-1}x^{\frac{14}{15}}+k=\\ \qquad\displaystyle \bbox[#FFECB3,5px]{\dfrac{8}{5}x^{\frac{5}{6}}-\dfrac{15}{14}x^{\frac{14}{15}}+k}$