What is the integral of cos^2(x)+2x+7 ?

The integral of cos^2(x)+2x+7 is x^2+15/2 x+1/4 sin(2x)+C

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integral of cos^2(x)+2x+7

\int cos^{2} (x) + 2 x + 7 d x

x^{2} + \frac{15}{2} x + \frac{1}{4} sin (2 x) + C

Solution steps

\int cos^{2} (x) + 2 x + 7 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^{2} (x) d x + \int 2 x d x + \int 7 d x

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

\int 2 x d x = x^{2}

\int 7 d x = 7 x

= \frac{1}{2} (x + \frac{1}{2} sin (2 x)) + x^{2} + 7 x

Simplify

\frac{1}{2} (x + \frac{1}{2} sin (2 x)) + x^{2} + 7 x : x^{2} + \frac{15}{2} x + \frac{1}{4} sin (2 x)

= x^{2} + \frac{15}{2} x + \frac{1}{4} sin (2 x)

Add a constant to the solution

= x^{2} + \frac{15}{2} x + \frac{1}{4} sin (2 x) + C

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What is the integral of cos^2(x)+2x+7 ?
The integral of cos^2(x)+2x+7 is x^2+15/2 x+1/4 sin(2x)+C