Evaluate the integral: I = integral dx/(cos^4x)

Evaluate the integral: $$ I = \int \frac{dx}{\cos^4x}. $$

Step 1 · Rewrite the integrand

The integrand is $\sec^4x$. Using $\sec^2x=1+\tan^2x$ on one factor:

$$ \sec^4x = \sec^2x\cdot\sec^2x = (1+\tan^2x)\sec^2x. $$

Thus, the integral becomes:

$$ I = \int (1+\tan^2x)\sec^2x\,dx. $$

Step 2 · Substitution

Let $u=\tan x$, so $du=\sec^2x\,dx$:

$$ I = \int (1+u^2)\,du. $$

Step 3 · Integrate

$$ I = u + \frac{u^3}{3} + C. $$

Step 4 · Substitute back

Substituting back $u=\tan x$:

$$ I = \tan x + \frac{\tan^3x}{3} + C. $$

Summary Table

Additional Notes

• The domain excludes $x=\tfrac\pi2+k\pi$ for integer $k$, where $\cos x=0$.
• This "split off $\sec^2x$, convert the rest to $\tan x$" strategy is the standard technique for any even power $\sec^{2n}x$, exactly mirroring the odd-power sine/cosine splitting in Problems 131–132.
• The result can be checked by differentiating $\tan x+\tfrac{\tan^3x}{3}$ and confirming it returns $\sec^4x=1/\cos^4x$.