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

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

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

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

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

Solution steps

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

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

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

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

Add a constant to the solution

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

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