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

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

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

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

ln ∣ tan (x) ∣ - ln ∣ cos (x) ∣ + C

Solution steps

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

Rewrite using trig identities

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

Expand

\frac{( 1 + sin ^{2} ( x ) ) sec ( x )}{sin ( x )} : \frac{sec ( x )}{sin ( x )} + sin (x) sec (x)

= \int \frac{sec ( x )}{sin ( x )} + sin (x) sec (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{sec ( x )}{sin ( x )} d x + \int sin (x) sec (x) d x

\int \frac{sec ( x )}{sin ( x )} d x = ln ∣ tan (x) ∣

\int sin (x) sec (x) d x = - ln ∣ cos (x) ∣

= ln ∣ tan (x) ∣ - ln ∣ cos (x) ∣

Add a constant to the solution

= ln ∣ tan (x) ∣ - ln ∣ cos (x) ∣ + C

\int (1 + u + u^{2} + u^{5}) d u

\int \frac{1}{x ln ( x ) + x} d x

\int (y^{2} - y^{3}) d y

\int - 4.15 x^{2} + 5.1 x + 6.096 d x

\int \frac{x ^{2} + x - 4}{x - 1} d x

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