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Question Number 129787 by EDWIN88 last updated on 19/Jan/21

$$\int x^7\sqrt{1+x^4}dx$$

Answered by bramlexs22 last updated on 19/Jan/21

Commented by MJS_new last updated on 19/Jan/21

$$\int x^7\sqrt{x^4+1}dx=$$$$[w=\sqrt{x^4+1}\to dx=\frac{\sqrt{x^4+1}}{2x^3}dw]$$$$=\frac{1}{2}\int (w^4-w^2)dw=\frac{w^5}{10}-\frac{w^3}{6}=\frac{1}{30}{w}^3(3w^2-5)=$$$$=\frac{{(x^4+1)}^{3/2}(3x^4-2)}{30}+C$$$$(=\frac{1}{30}(3x^8+x^4-2)\sqrt{x^4+1}+C)$$

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