Question

Please solve sec^8x/cosec x .dx

Answer 1

∫sec8(x)cosec(x)dx∫sec8(x)cosec(x)dx

Since sec(x)=1cos(x)sec(x)=1cos(x)

And cosex(x)=1sin(x)cosex(x)=1sin(x)

The equation can be written as

∫sin(x)cos(x)sec7(x)dx∫sin(x)cos(x)sec7(x)dx

∫tan(x)sec7(x)dx∫tan(x)sec7(x)dx

Let u=sec(x)u=sec(x)

dudx=sec(x)tan(x)dxdudx=sec(x)tan(x)dx

Substituting,

∫tan(x)sec7(x)dx=∫u6du∫tan(x)sec7(x)dx=∫u6du

Integrating,

=[u77]=[u77]

Substituting u back,

=sec7(x)7+C...