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

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

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

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

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

Solution steps

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

Use the basic trigonometric identity:

\frac{1}{cos ( x )} = sec (x)

= \int sec^{2} (x) (3 + sin (x)) d x

Apply Integration By Parts

: - \int sin (x) d x + tan (x) (3 + sin (x))

= - \int sin (x) d x + tan (x) (3 + sin (x))

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

= - (- cos (x)) + tan (x) (3 + sin (x))

Simplify

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

Add a constant to the solution

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

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