How do you integrate $\int {cos}^{3} θ \sqrt{sin θ}$ $int cos^3thetasqrt(sintheta)$ ?

Question

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Answer 1 · 2017-10-19T19:36:16

$= \frac{2}{3} {sin}^{\frac{3}{2}} x - \frac{2}{7} {sin}^{\frac{7}{2}} x + C$ $=2/3sin^(3/2)x-2/7sin^(7/2)x+C$

for $θ$ $theta$ read $x$ $x$

$\int {cos}^{3} x {sin}^{\frac{1}{2}} x d x$ $intcos^3xsin^(1/2)xdx$

$= \int cos x {cos}^{2} x {sin}^{\frac{1}{2}} x d x$ $=intcosxcos^2xsin^(1/2 )xdx$

$= \int cos x (1 - {sin}^{2} x) {sin}^{\frac{1}{2}} x d x$ $=intcosx(1-sin^2x)sin^(1/2)xdx$

$= \int cos x {sin}^{\frac{1}{2}} x - cos x {sin}^{\frac{5}{2}} x d x$ $=intcosxsin^(1/2)x-cosxsin^(5/2)xdx$

by inspection

$= \frac{2}{3} {sin}^{\frac{3}{2}} x - \frac{2}{7} {sin}^{\frac{7}{2}} x + C$ $=2/3sin^(3/2)x-2/7sin^(7/2)x+C$

which can be simplified as required

How do you integrate int cos^3thetasqrt(sintheta)∫cos3θ√sinθint cos^3thetasqrt(sintheta)?