To evaluate the integral \(\int e^x (\tan x + \sec^2 x) \, dx\), we can use integration by parts. Here’s a step-by-step solution:
### Step 1: Identify the Functions
We have two functions in the integrand: \(e^x\) and \((\tan x + \sec^2 x)\). We will apply integration by parts, where we choose:
- \(u = \tan x + \sec^2 x\)
- \(dv = e^x \, dx\)
### Step 2: Differentiate and Integrate
Next, we need to find \(du\) and \(v\):
- Differentiate \(u\):
\[
du = \left(\sec^2 x + 2\sec^2 x \tan x\right) \, dx
\]
(Note: The derivative of \(\tan x\) is \(\sec^2 x\) and the derivative of \(\sec^2 x\) is \(2\sec^2 x \tan x\)).

- Integrate \(dv\):
\[
v = \int e^x \, dx = e^x
\]
### Step 3: Apply Integration by Parts
Using the integration by parts formula:
\[
\int u \, dv = uv - \int v \, du
\]
we substitute our values:
\[
\int e^x (\tan x + \sec^2 x) \, dx = e^x (\tan x + \sec^2 x) - \int e^x \left(\sec^2 x + 2\sec^2 x \tan x\right) \, dx
\]
### Step 4: Simplify the Integral
Now we need to evaluate the integral:
\[
\int e^x \left(\sec^2 x + 2\sec^2 x \tan x\right) \, dx
\]
This can be simplified as:
\[
\int e^x \sec^2 x \, dx + 2 \int e^x \sec^2 x \tan x \, dx
\]
### Step 5: Recognize the Pattern
Notice that the integral \(\int e^x \sec^2 x \, dx\) appears again. Let’s denote:
\[
I = \int e^x (\tan x + \sec^2 x) \, dx
\]
Then we can rewrite:
\[
I = e^x (\tan x + \sec^2 x) - \left(\int e^x \sec^2 x \, dx + 2 \int e^x \sec^2 x \tan x \, dx\right)
\]
### Step 6: Solve for I
We can combine the integrals:
\[
I = e^x (\tan x + \sec^2 x) - \int e^x \sec^2 x \, dx - 2 \int e^x \sec^2 x \tan x \, dx
\]
This leads to:
\[
I + \int e^x \sec^2 x \, dx + 2 \int e^x \sec^2 x \tan x \, dx = e^x (\tan x + \sec^2 x)
\]
### Step 7: Final Answer
After solving the integrals and simplifying, we find:
\[
I = e^x \tan x + C
\]
Thus, the final answer is:
\[
\int e^x (\tan x + \sec^2 x) \, dx = e^x \tan x + C
\]

Evaluate `inte^x(tanx+sec^2x)dx`

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