What is the integral of x^2-e^{2x}+cos(x)+3/(1+x^2) ?

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

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integral of x^2-e^{2x}+cos(x)+3/(1+x^2)

\int x^{2} - e^{2 x} + cos (x) + \frac{3}{1 + x ^{2}} d x

\frac{x ^{3}}{3} - \frac{1}{2} e^{2 x} + sin (x) + 3 arctan (x) + C

Solution steps

\int x^{2} - e^{2 x} + cos (x) + \frac{3}{1 + x ^{2}} 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 x^{2} d x - \int e^{2 x} d x + \int cos (x) d x + \int \frac{3}{1 + x ^{2}} d x

\int x^{2} d x = \frac{x ^{3}}{3}

\int e^{2 x} d x = \frac{1}{2} e^{2 x}

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

\int \frac{3}{1 + x ^{2}} d x = 3 arctan (x)

= \frac{x ^{3}}{3} - \frac{1}{2} e^{2 x} + sin (x) + 3 arctan (x)

Add a constant to the solution

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

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