Evaluate the integral. ∫tan5xdx

Question

Evaluate the integral.  ∫tan5xdx

Alex Sheppard · Accepted Answer

Step 1  Split tan5x into simpler powers tan3x and tan2x and change tan2x to sec2x−1 and simplify.  ∫tan5xdx=∫tan3x(tan2xdx)  =∫tan3x(sec2x−1)dx  =∫(tan3xsec2x−tan3x)dx  Step 2  use the integral theorem and integrate both the of ∫(tan3xsec2x−tan3x)dx terms with respect to dx and simplify to obtain the value of the integral.  ∫(tan3xsec2x−tan3x)dx=∫tan3xsec2xdx−∫tan3xdx  =∫(tan3xsec2x)dx−∫tan2xdx(tan⁡x)dx  =∫tan3xsec2xdx−∫(sec2x−1)tan⁡xdx  =∫tan3xsec2xdx−∫tan⁡xsec2xdx+∫tan⁡xdx  =14tan4x−12tan2x+ln⁡sec⁡x+C

Linda Birchfield · Accepted Answer

∫tan⁡(x)5dx  Evaluate the integral  14×tan⁡(x)4−∫tan⁡(x)3dx  Evaluate the integral  14×tan⁡(x)4−(12×tan⁡(x)2−∫tan⁡(x)dx)  Expand the expression  14×tan⁡(x)4−(12×tan⁡(x)2−∫sin⁡(x)cos⁡(x)dx)  Transform the expression  14×tan⁡(x)4−(12×tan⁡(x)2−∫−1tdt)  Use properties of integrals  14×tan⁡(x)4−(12×tan⁡(x)2+∫1tdt)  Evaluate the integral  14×tan⁡(x)4−(12×tan⁡(x)2+ln⁡(|t|))  Substitute back  14×tan⁡(x)4−(12×tan⁡(x)2+ln⁡(|cos⁡(x)|))  Simplify  tan⁡(x)44−tan⁡(x)22−ln⁡(|cos⁡(x)|)  Add C  Solution  tan⁡(x)44−tan⁡(x)22−ln⁡(|cos⁡(x)|)+C

karton · Accepted Answer

tan5⁡(x)dx=∫(tan2⁡(x))2tan⁡(x)dx=∫(sec2⁡(x)−1)2tan⁡(x)dx=∫(u2−1)2udu=∫(u3−2u+1u)du=∫u3du−2∫udu+∫1udu∫u3du=u44∫udu=u22∫1udu=ln⁡(u)∫u3du−2∫udu+∫1udu=ln⁡(u)+u44−u2=ln⁡(sec⁡(x))+sec4⁡(x)4−sec2⁡(x)∫tan5⁡(x)dx=ln⁡(|sec⁡(x)|)+sec4⁡(x)4−sec2⁡(x)+C

Evaluate the integral. \int \tan^{5}xdx

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