#### To determine

**To find:**

The general indefinite integral and illustrate by graphing several members of the family on the same screen.

#### Answer

sinx+x24+C

And the graph of the family of the curve is

#### Explanation

**1) Concept:**

First, by using rules of indefinite integrals evaluate the given integral by taking several values of C. Graph the curves on the same screen.

**2) Formula:**

i. ∫fx+gxdx=∫f(x)dx+∫g(x)dx

ii. ∫xndx=xn+1n+1+C

iii. ∫cos xdx= sinx+C

iv. ∫cf(x)dx=c∫f(x)dx

**3) Given:**

∫cos x+12xdx

**4) Calculation:**

Consider, ∫cos x+12xdx integrating by using rules of integration.

Separating the integral by using the rule, ∫fx+gxdx=∫f(x)dx+∫g(x)dx,

∫cos x+12xdx= ∫cosxdx+∫12xdx

By taking the constant outside the integral,

= ∫cosxdx+12∫xdx

By applying the power rule of integration,

=sinx+12·x22+C

=sinx+x24+C

Therefore,

∫cos x+12xdx= sinx+x24+C

By taking the values of c as c= -10, -5, 0, 5 and 10 and plotting graph on same screen,

**Conclusion:**

Therefore,

∫cos x+12xdx= sinx+x24+C

And the graph of the family of the curve is