Question Statement

Evaluate the integral:

$\int e^y\sin e^{y}dy$

---

Background and Explanation

This problem requires the method of integration by substitution (also known as u-substitution). The key insight is recognizing that the integrand contains a composite function $\sin (e^y)$ multiplied by the derivative of its inner function $e^y$, which allows for a clean substitution.

---

Solution

Let $I$ denote the integral we wish to evaluate:

$I=\int e^y\sin e^{y}dy$

We can rearrange the integrand to group terms strategically, making the substitution pattern more apparent:

$I=\int \left(\sin e^y\right)\cdot e^{y}dy$

Notice that the derivative of $e^y$ is $e^y$ itself. This suggests we should substitute the inner function $e^y$ with a new variable.

Step 1: Perform the substitution

Put: $e^y=t$

Differentiating both sides with respect to $y$: $\frac{d}{dy}(e^y)=\frac{dt}{dy}$

$e^y=\frac{dt}{dy}$

Therefore: $e^{y}dy=dt$

Step 2: Transform the integral

Substituting $t=e^y$ and $e^{y}dy=dt$ into our integral:

$I=\int \sin tdt$

Step 3: Integrate with respect to $t$

The integral of sine is standard: $I=-\cos t+c$

where $c$ is the constant of integration.

Step 4: Back-substitute to return to the original variable

Replace $t$ with $e^y$ to express the answer in terms of the original variable $y$:

$I=-\cos \left(e^y\right)+c$

Thus, the final solution is:

$\int e^y\sin e^{y}dy=-\cos \left(e^y\right)+c$

---

Key Formulas or Methods Used

• Integration by Substitution: $\int f(g(x))\cdot g^{\prime }(x)dx=\int f(u)du$ where $u=g(x)$
• Standard Integral: $\int \sin xdx=-\cos x+C$
• Derivative Rule: $\frac{d}{dy}(e^y)=e^y$ (which enables the substitution)

---

Summary of Steps

1. Identify that $e^y$ appears both as the argument of sine and as a multiplier (which is the derivative of $e^y$)
2. Substitute $t=e^y$, which implies $e^{y}dy=dt$
3. Transform the integral to $\int \sin tdt$
4. Integrate to get $-\cos t+c$
5. Substitute back $t=e^y$ to obtain the final answer $-\cos (e^y)+c$