What is the integral of ((3+cos^3(x)))/(cos^2(x)) ?

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

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integral of ((3+cos^3(x)))/(cos^2(x))

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

3 tan (x) + sin (x) + C

Solution steps

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

Expand

\frac{3 + cos ^{3} ( x )}{cos ^{2} ( x )} : \frac{3}{cos ^{2} ( x )} + cos (x)

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

\int \frac{3}{cos ^{2} ( x )} d x = 3 tan (x)

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

= 3 tan (x) + sin (x)

Add a constant to the solution

= 3 tan (x) + sin (x) + C

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