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Question Number 53382 by gunawan last updated on 21/Jan/19

$$\int {\cos }^{3}x{e}^{\log \left(\sin x\right)}dx=$$

Answered by math1967 last updated on 21/Jan/19

$$\int (1-{sin}^{2}x){cosxe}^{ln\left(sinx\right)}dx$$$$letsinx=z\therefore cosxdx=dz$$$$\int (1-z^2)e^{lnz}dz=\int (1-z^2)zdz$$$$\int zdz-\int z^{3}dz=\frac{1}{2}{z}^2-\frac{1}{4}{z}^4+c$$$$=\frac{1}{2}\sin {}^{2}x-\frac{1}{4}{sin}^{4}x+cans$$

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