What is the integral of 3/(1+x^2)+4sin(x)+1 ?

The integral of 3/(1+x^2)+4sin(x)+1 is 3arctan(x)-4cos(x)+x+C

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integral of 3/(1+x^2)+4sin(x)+1

\int \frac{3}{1 + x ^{2}} + 4 sin (x) + 1 d x

3 arctan (x) - 4 cos (x) + x + C

Solution steps

\int \frac{3}{1 + x ^{2}} + 4 sin (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 \frac{3}{1 + x ^{2}} d x + \int 4 sin (x) d x + \int 1 d x

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

\int 4 sin (x) d x = - 4 cos (x)

\int 1 d x = x

= 3 arctan (x) - 4 cos (x) + x

Add a constant to the solution

= 3 arctan (x) - 4 cos (x) + x + C

What is the integral of 3/(1+x^2)+4sin(x)+1 ?
The integral of 3/(1+x^2)+4sin(x)+1 is 3arctan(x)-4cos(x)+x+C