What is the integral of tan(x)(tan(x)+1) ?

The integral of tan(x)(tan(x)+1) is tan(x)-x-ln|cos(x)|+C

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integral of tan(x)(tan(x)+1)

\int tan (x) (tan (x) + 1) d x

tan (x) - x - ln ∣ cos (x) ∣ + C

Solution steps

\int tan (x) (tan (x) + 1) d x

Rewrite using trig identities

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

Expand

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

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

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

\int \frac{sin ( x )}{cos ( x )} d x = - ln ∣ cos (x) ∣

= tan (x) - x - ln ∣ cos (x) ∣

Add a constant to the solution

= tan (x) - x - ln ∣ cos (x) ∣ + C

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