Evaluate the integral.

$\int {\mathrm{sec}}^{3}x{\mathrm{tan}}^{4}xdx$

Dolly Robinson
2021-10-20
Answered

Evaluate the integral.

$\int {\mathrm{sec}}^{3}x{\mathrm{tan}}^{4}xdx$

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tabuordg

Answered 2021-10-21
Author has **99** answers

The given integral is:

$I=\int {\mathrm{sec}}^{3}x{\mathrm{tan}}^{4}xdx$

$=\int {\mathrm{sec}}^{3}x{\left({\mathrm{tan}}^{2}x\right)}^{2}dx$

$=\int {\mathrm{sec}}^{3}x{({\mathrm{sec}}^{2}x-1)}^{2}dx$

$=\int {\mathrm{sec}}^{3}x({\mathrm{sec}}^{4}x-2{\mathrm{sec}}^{2}x+1)dx$

$=\int {\mathrm{sec}}^{7}xdx-2\int {\mathrm{sec}}^{5}xdx+\int {\mathrm{sec}}^{3}xdx$

Solving each of the component integrals we get,

$I=\frac{\mathrm{ln}(\mathrm{tan}x+\mathrm{sec}x)}{16}+\frac{{\mathrm{sec}}^{5}x\mathrm{tan}x}{6}-\frac{7{\mathrm{sec}}^{3}x\mathrm{tan}x}{24}+\frac{\mathrm{sec}x\mathrm{tan}x}{16}+C$

where C is the constant of integration.

Solving each of the component integrals we get,

where C is the constant of integration.

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