What is the integral of cos^3(x)(1+sqrt(sin(x))) ?

The integral of cos^3(x)(1+sqrt(sin(x))) is sin(x)-(sin^3(x))/3+2/3 sin^{3/2}(x)-2/7 sin^{7/2}(x)+C

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integral of cos^3(x)(1+sqrt(sin(x)))

\int cos^{3} (x) (1 + sin (x)) d x

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

Solution steps

\int cos^{3} (x) (1 + sin (x)) d x

Expand

cos^{3} (x) (1 + sin (x)) : cos^{3} (x) + cos^{3} (x) sin (x)

= \int cos^{3} (x) + cos^{3} (x) sin (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int cos^{3} (x) d x + \int cos^{3} (x) sin (x) d x

\int cos^{3} (x) d x = sin (x) - \frac{sin ^{3} ( x )}{3}

\int cos^{3} (x) sin (x) d x = \frac{2}{3} sin^{\frac{3}{2}} (x) - \frac{2}{7} sin^{\frac{7}{2}} (x)

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

Add a constant to the solution

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

\frac{d y}{d x}, ln (x y) + 9 x = 40

\int \frac{16 y - 9}{3 y - 5 y ^{2}} d y

\int x^{1} cos (- 9 x) d x

x \to 3 lim (\frac{∣ x - 2 ∣}{x - 2})

y = \frac{4 x ^{7}}{7}

What is the integral of cos^3(x)(1+sqrt(sin(x))) ?
The integral of cos^3(x)(1+sqrt(sin(x))) is sin(x)-(sin^3(x))/3+2/3 sin^{3/2}(x)-2/7 sin^{7/2}(x)+C