Evaluate the indefinite integral. (Use C for the constant of integration.) Integral of sin(2x)...

Question:

Evaluate the indefinite integral. (Use {eq}C {/eq} for the constant of integration.)

{eq}\int \sin(2x) \sqrt{\cos(2x) + 8} \, \mathrm{d}x {/eq}

Answer and Explanation:


Given integral is :

{eq}\int \sin(2x) \sqrt{\cos(2x) + 8} \, \mathrm{d}x\\ \mathrm{Apply\:u-substitution:}\:u=2x\\ =\frac{1}{2}\cdot \int \:\sin \left(u\right)\sqrt{\cos \left(u\right)+8}du\\ \mathrm{Apply\:u-substitution:}\:v=\cos \left(u\right)+8\\ =\frac{1}{2}\cdot \int \:-\sqrt{v}dv\\ \mathrm{Apply\:the\:Power\:Rule}:\quad \int x^adx=\frac{x^{a+1}}{a+1},\:\quad \:a\ne -1\\ =\frac{1}{2}\left(-\frac{v^{\frac{1}{2}+1}}{\frac{1}{2}+1}\right)+c\\ \mathrm{Substitute\:back}\:v=\cos \left(u\right)+8,\:u=2x\\ =\frac{1}{2}\left(-\frac{\left(\cos \left(2x\right)+8\right)^{\frac{1}{2}+1}}{\frac{1}{2}+1}\right)+c\\ =-\frac{1}{3}\left(\cos \left(2x\right)+8\right)^{\frac{3}{2}}+C\\ {/eq}


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