Question

$\int \frac{\left(1+x\right)e^x}{{\sin }^2\left(x{e}^x\right)}dx=$

• A. $-cot{e}^x+C$
• B. $\tan \left(x{e}^x\right)+C$
• C. $\tan \left(e^x\right)+C$
• D. $cot\left(x{e}^x\right)+C$
• E. $-cot\left(x{e}^x\right)+C$

Answer 1

The correct option is E

$-cot\left(x{e}^x\right)+C$

Explanation for Correct answer:

Finding the value of the given integral:

The given equation is $\int \frac{\left(1+x\right)e^x}{{\sin }^2\left(x{e}^x\right)}dx$

Solving this by using the substitution method

$$\begin{gathered} t=x{e}^x \\ dt=\left(x{e}^x+1.e^x\right)dx\left[\because \frac{d}{dx}uv=u.dv+v.du\right] \end{gathered}$$

$$\begin{gathered} \int \frac{\left(1+x\right)e^x}{{\sin }^2\left(x{e}^x\right)}dx=\int \frac{dt}{{\sin }^2\left(t\right)} \\ =\int \cos e{c}^2\left(t\right)dt \\ =-cott+C \\ =-cot\left(x{e}^x\right)+C \end{gathered}$$

Hence, option (E) is the correct answer.