What is the integral of (sqrt(sin(x))+cos(x))^2 ?

The integral of (sqrt(sin(x))+cos(x))^2 is -cos(x)+4/3 sin^{3/2}(x)+1/2 (x+1/2 sin(2x))+C

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integral of (sqrt(sin(x))+cos(x))^2

\int (sin (x) + cos (x))^{2} d x

- cos (x) + \frac{4}{3} sin^{\frac{3}{2}} (x) + \frac{1}{2} (x + \frac{1}{2} sin (2 x)) + C

Solution steps

\int (sin (x) + cos (x))^{2} d x

Expand

(sin (x) + cos (x))^{2} : sin (x) + 2 cos (x) sin (x) + cos^{2} (x)

= \int sin (x) + 2 cos (x) sin (x) + cos^{2} (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 sin (x) d x + \int 2 cos (x) sin (x) d x + \int cos^{2} (x) d x

\int sin (x) d x = - cos (x)

\int 2 cos (x) sin (x) d x = \frac{4}{3} sin^{\frac{3}{2}} (x)

\int cos^{2} (x) d x = \frac{1}{2} (x + \frac{1}{2} sin (2 x))

= - cos (x) + \frac{4}{3} sin^{\frac{3}{2}} (x) + \frac{1}{2} (x + \frac{1}{2} sin (2 x))

Add a constant to the solution

= - cos (x) + \frac{4}{3} sin^{\frac{3}{2}} (x) + \frac{1}{2} (x + \frac{1}{2} sin (2 x)) + C

\int 2 x csc^{2} (6) x^{2} d x

\int (2 cos (x) - sin (3 x) cos (3 x)) d x

\int (3 x^{2} - 2 x) e^{2 x} d x

\int \frac{z ^{2} + 2 z + 3}{( z - 6 ) ( z ^{2} + 4 )} d z

\int \frac{tan ^{2} ( x )}{sin ^{2} ( x )} d x

What is the integral of (sqrt(sin(x))+cos(x))^2 ?
The integral of (sqrt(sin(x))+cos(x))^2 is -cos(x)+4/3 sin^{3/2}(x)+1/2 (x+1/2 sin(2x))+C