Question

Evaluate the integral ∫tan³xsec²x dx

Answer 1

See the image. Hope this helps!

Answer 2

`(1)/(4)\tan^4 x+C`

Explanation:

To solve the given integral, we'll use the substitution method to simplify the integration process.

Given integral:

`\displaystyle \int (\tan^3{x}\sec^2x)dx`

`\hrulefill`

The integral of tan³xsec²x can be approached by recognizing that sec²x is the derivative of tanx, which suggests the use of u-substitution where u = tanx. This way, du = sec²x dx and the integral becomes:

`u =\tan x \rightarrow du=\sec^2x dx\\\\\\\\\Longrightarrow \displaystyle \int (u^3\sec^2x)(du)/(\sec^2x) \\\\\\\\\Longrightarrow \displaystyle \int u^3du`

This is a straightforward power rule integral:

`\Longrightarrow (u^(3+1))/(3+1) +C\\\\\\\\\Longrightarrow (u^(4))/(4)+C \\\\\\\\\Longrightarrow (1)/(4)u^4+C`

Substituting back in 'u':

`\Longrightarrow (1)/(4)(\tan x)^4+C\\\\\\\\\therefore \displaystyle \int (\tan^3{x}\sec^2x)dx= \boxed{(1)/(4)\tan^4 x+C}`

Thus, the integral is found. Where 'C' is the constant of integration.