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

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

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integral of (cos^3(x)+1)/(cos^2(x))

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

sin (x) + tan (x) + C

Solution steps

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

Expand

\frac{cos ^{3} ( x ) + 1}{cos ^{2} ( x )} : cos (x) + \frac{1}{cos ^{2} ( x )}

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

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

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

= sin (x) + tan (x)

Add a constant to the solution

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

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