What is the integral of cos^2(x)+4x^3+sin^2(x) ?

The integral of cos^2(x)+4x^3+sin^2(x) is x^4+x+C

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integral of cos^2(x)+4x^3+sin^2(x)

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

x^{4} + x + C

Solution steps

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

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

= \int 4 x^{3} + 1 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 4 x^{3} d x + \int 1 d x

\int 4 x^{3} d x = x^{4}

\int 1 d x = x

= x^{4} + x

Add a constant to the solution

= x^{4} + x + C

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What is the integral of cos^2(x)+4x^3+sin^2(x) ?
The integral of cos^2(x)+4x^3+sin^2(x) is x^4+x+C