What is the integral of 2sin(x)+2x^2+5x+1 ?

The integral of 2sin(x)+2x^2+5x+1 is -2cos(x)+(2x^3)/3+(5x^2)/2+x+C

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integral of 2sin(x)+2x^2+5x+1

\int 2 sin (x) + 2 x^{2} + 5 x + 1 d x

- 2 cos (x) + \frac{2 x ^{3}}{3} + \frac{5 x ^{2}}{2} + x + C

Solution steps

\int 2 sin (x) + 2 x^{2} + 5 x + 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 2 sin (x) d x + \int 2 x^{2} d x + \int 5 x d x + \int 1 d x

\int 2 sin (x) d x = - 2 cos (x)

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

\int 5 x d x = \frac{5 x ^{2}}{2}

\int 1 d x = x

= - 2 cos (x) + \frac{2 x ^{3}}{3} + \frac{5 x ^{2}}{2} + x

Add a constant to the solution

= - 2 cos (x) + \frac{2 x ^{3}}{3} + \frac{5 x ^{2}}{2} + x + C

\frac{d}{d x} (\frac{2 x - 3}{x ^{2}})

\int \frac{81 + x ^{2}}{x} d x

x \to \infty lim (3 x^{3} + 3 x^{2} - x)

\frac{1}{s ^{2} ( s ^{2} + 1 )}

\int \frac{6 - x}{6 + x} d x

What is the integral of 2sin(x)+2x^2+5x+1 ?
The integral of 2sin(x)+2x^2+5x+1 is -2cos(x)+(2x^3)/3+(5x^2)/2+x+C