$\displaystyle \displaystyle \displaystyle\int{{{x}^{2}}{{({{x}^{3}}+3)}^{3}}}\,\mathrm{d}x$ \(\di

Question

$\displaystyle \displaystyle \displaystyle\int{{{x}^{2}}{{({{x}^{3}}+3)}^{3}}}\,\mathrm{d}x$ $\displaystyle \displaystyle \displaystyle\int{(5{{x}^{4}}+4x)({{x}^{5}}+2{{x}^{2}}}+1{{)}^{3}}\,\mathrm{d}x$ $\displaystyle \displaystyle \displaystyle\int{\frac{3x-2}{{{\left( 3{{x}^{2}}-4x-1 \right)}^{3}}}}\,\mathrm{d}x$ $\displaystyle \displaystyle \displaystyle\int{\frac{{{x}^{2}}}{\sqrt{{{x}^{3}}+1}}\,\mathrm{d}x}$ $\displaystyle \displaystyle \displaystyle\int{\frac{21{{x}^{2}}+14}{\sqrt{{{x}^{3}}+2x+1}}\,\mathrm{d}x}$ $\displaystyle \displaystyle \displaystyle\int{\frac{x}{{{\sqrt{1-2{{x}^{2}}}}^{5}}}\,\mathrm{d}x}$ $\displaystyle \displaystyle \displaystyle\int{{{\left( \frac{x}{\sqrt{3{{x}^{4}}-1}} \right)}^{3}}\,\mathrm{d}x}$ $\displaystyle \displaystyle \displaystyle\int{3{{x}^{3}}{{(2{{x}^{4}}-5)}^{8}}}\,\mathrm{d}x$ $\displaystyle \displaystyle \displaystyle\int{{{e}^{3x}}{{({{e}^{3x}}+5)}^{11}}}\,\mathrm{d}x$ $\displaystyle \displaystyle \displaystyle\int{\frac{{{x}^{3}}}{{{x}^{4}}+3}\text{}\hspace{0.5em}}\mathrm{d}x$

Accepted Answer

(a) $\dfrac1{12}(x^3+3)^4+C$ (b) $\dfrac14(x^5+2x^2+1)^4+C$ (c) $-\dfrac1{4(3x^2-4x-1)^2}+C$ (d) $\dfrac23\sqrt{x^3+1}+C$ (e) $14\sqrt{x^3+2x+1}+C$ (f) $\dfrac1{6(1-2x^2)^{3/2}}+C$ (g) $-\dfrac1{6\sqrt{3x^4-1}}+C$ (h) $\dfrac1{24}(2x^4-5)^9+C$ (i) $\dfrac1{36}(\mathrm e^{3x}+5)^{12}+C$ (j) $\dfrac14\ln|x^4+3|+C$

Techniques of Integration - Integration by the Reverse Chain Rule

Techniques of Integration - Integration by the Reverse Chain Rule

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