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

The integral of (sin(x)+cos(x))^3 is -cos(x)+(cos^3(x))/3+sin^3(x)-cos^3(x)+sin(x)-(sin^3(x))/3+C

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

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

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

Solution steps

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

Expand

(sin (x) + cos (x))^{3} : sin^{3} (x) + 3 sin^{2} (x) cos (x) + 3 cos^{2} (x) sin (x) + cos^{3} (x)

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

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

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

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

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

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

Add a constant to the solution

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

What is the integral of (sin(x)+cos(x))^3 ?
The integral of (sin(x)+cos(x))^3 is -cos(x)+(cos^3(x))/3+sin^3(x)-cos^3(x)+sin(x)-(sin^3(x))/3+C