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

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

integral of 1+cos^2(x)

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

\frac{3}{2} x + \frac{1}{4} sin (2 x) + C

Solution steps

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

\int 1 d x = x

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

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

Simplify

x + \frac{1}{2} (x + \frac{1}{2} sin (2 x)) : \frac{3}{2} x + \frac{1}{4} sin (2 x)

= \frac{3}{2} x + \frac{1}{4} sin (2 x)

Add a constant to the solution

= \frac{3}{2} x + \frac{1}{4} sin (2 x) + C

\int 12 x^{2} - 6 x + 1 d x

y^{^{''}} - 3 y^{^{'}} = 9 t, y (0) = 3, y^{^{'}} (0) = 0

im pl i c i t \frac{x}{y} + \frac{y}{x} = 3 πa b^{2}

in v erse l a pl a ce \frac{( - 2 s - 1 )}{s + 1}

\int \frac{1}{y ^{2} + x ^{2}} d x

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