Evaluate the integral: I = integral dx/(cos x)
Evaluate the integral: $$ I = \int \frac{dx}{\cos x}. $$
Explanation of Method
This is the classical secant integral. Multiplying the integrand by $\dfrac{\sec x+\tan x}{\sec x+\tan x}$ turns the numerator into the derivative of the denominator, producing a standard logarithmic antiderivative.
- Rewrite $1/\cos x$ as $\sec x$.
- Multiply by the factor $(\sec x+\tan x)/(\sec x+\tan x)$.
- Recognize the numerator as the derivative of the denominator.
- Integrate using the standard logarithmic result.
Step 1 · Rewrite the integrand
The integrand $\dfrac{1}{\cos x}$ is the secant function:
$$ I = \int \sec x\,dx. $$
Step 2 · Apply the standard technique
Multiplying and dividing by $\sec x+\tan x$:
$$ I = \int \sec x\cdot\frac{\sec x+\tan x}{\sec x+\tan x}\,dx = \int \frac{\sec^2x+\sec x\tan x}{\sec x+\tan x}\,dx. $$
The numerator is exactly the derivative of the denominator, since $\dfrac{d}{dx}(\sec x+\tan x)=\sec x\tan x+\sec^2x$.
Step 3 · Integrate
Letting $w=\sec x+\tan x$, so $dw=(\sec x\tan x+\sec^2x)\,dx$:
$$ I = \int \frac{dw}{w} = \ln|w| + C. $$
Step 4 · Substitute back
Substituting back $w=\sec x+\tan x$:
$$ I = \ln|\sec x+\tan x| + C. $$
Summary Table
| Quantity | Value |
|---|---|
| Trick | multiply by $(\sec x+\tan x)/(\sec x+\tan x)$ |
| Substitution | $w = \sec x + \tan x$ |
| Antiderivative | $\ln|w|$ |
Additional Notes
- The domain excludes $x=\tfrac\pi2+k\pi$ for integer $k$, where $\cos x=0$.
- This is the counterpart of Problem 112's cosecant integral, using $\sec x+\tan x$ in place of $\csc x-\cot x$; the same "multiply by a clever form of 1" strategy applies.
- The result can be checked by differentiating $\ln|\sec x+\tan x|$ with the chain rule and confirming it returns $\sec x = 1/\cos x$.
Result
The solution to the integral is: