Question

∫tan(x)³dxplease someone solve this for me ​

Answer 1

Answer:

½ sec²(x) + ln(|cos(x)|) + C

Step-by-step explanation:

∫ tan³(x) dx

∫ tan²(x) tan(x) dx

∫ (sec²(x) − 1) tan(x) dx

∫ (sec²(x) tan(x) − tan(x)) dx

∫ sec²(x) tan(x) dx − ∫ tan(x) dx

For the first integral, if u = sec(x), then du = sec(x) tan(x) dx.

∫ u du = ½ u² + C

Substituting back:

½ sec²(x) + C

For the second integral, tan(x) = sin(x) / cos(x).  If u = cos(x), then du = -sin(x) dx.

∫ -du / u = -ln(u) + C

Substituting back:

-ln(|cos(x)|) + C

Therefore, the total integral is:

½ sec²(x) + ln(|cos(x)|) + C