Question

Evaluate ∫(2tanxcosx+7) dx. Here C is the constant of integration.

Answer 1

Given:

The function is:

`\int(2\tan x\cos x+7)dx`

Find-:

Evaluate the expression

Explanation-:

Use integration property is:

`\int(a+b)dx=\int adx+\int bdx`

So the expression is:

`\begin{gathered} =\int(2\tan x\cos x+7)dx \\ \\ =\int2\tan x\cos xdx+\int7dx \\ \\ =2\int\tan(x)\cos(x)dx+7\int1dx \end{gathered}`

Simplify the expression is:

`\begin{gathered} =2\int\tan(x)\cos(x)+7\int1dx \\ \\ =2\int(\sin x)/(\cos x)*\cos(x)dx+7\int1dx \\ \\ =2\int\sin(x)dx+7\int1dx \end{gathered}`

Use integration formula is:

`\begin{gathered} \int\sin x=-\cos x \\ \\ \int1dx=x \end{gathered}`

So, the value is:

`\begin{gathered} =2\int\sin(x)dx+7\int1dx \\ \\ =-2\cos x+7x+C \end{gathered}`