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

Holly Guerrero 2021-12-31 Answered
Evaluate the integral.
tan5xdx
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Expert Answer

Alex Sheppard
Answered 2022-01-01 Author has 36 answers
Step 1
Split tan5x into simpler powers tan3x and tan2x and change tan2x to sec2x1 and simplify.
tan5xdx=tan3x(tan2xdx)
=tan3x(sec2x1)dx
=(tan3xsec2xtan3x)dx
Step 2
use the integral theorem and integrate both the of (tan3xsec2xtan3x)dx terms with respect to dx and simplify to obtain the value of the integral.
(tan3xsec2xtan3x)dx=tan3xsec2xdxtan3xdx
=(tan3xsec2x)dxtan2xdx(tanx)dx
=tan3xsec2xdx(sec2x1)tanxdx
=tan3xsec2xdxtanxsec2xdx+tanxdx
=14tan4x12tan2x+lnsecx+C
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Linda Birchfield
Answered 2022-01-02 Author has 39 answers
tan(x)5dx
Evaluate the integral
14×tan(x)4tan(x)3dx
Evaluate the integral
14×tan(x)4(12×tan(x)2tan(x)dx)
Expand the expression
14×tan(x)4(12×tan(x)2sin(x)cos(x)dx)
Transform the expression
14×tan(x)4(12×tan(x)21tdt)
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)44tan(x)22ln(|cos(x)|)
Add C
Solution
tan(x)44tan(x)22ln(|cos(x)|)+C
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karton
Answered 2022-01-04 Author has 368 answers

tan5(x)dx=(tan2(x))2tan(x)dx=(sec2(x)1)2tan(x)dx=(u21)2udu=(u32u+1u)du=u3du2udu+1uduu3du=u44udu=u221udu=ln(u)u3du2udu+1udu=ln(u)+u44u2=ln(sec(x))+sec4(x)4sec2(x)tan5(x)dx=ln(|sec(x)|)+sec4(x)4sec2(x)+C

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