Evaluate the definite integral:

$\int_0^{\pi/4} an x \ dx$

Question

Evaluate the definite integral:

$\int_0^{\pi/4} 	an x \ dx$

Accepted Answer

> [!TLDR]
> The integral $\int_0^{\pi/4} 	an x \, dx$ is solved by rewriting $	an x$ as $\frac{\sin x}{\cos x}$ and using the substitution $u = \cos x$, which transforms the integral into a standard logarithmic form. The final value is $\frac{1}{2} \log 2$.

## Why This Approach Works

The integral of $	an x$ is a classic example where direct integration isn't obvious, but a simple substitution makes it clean. The key insight is that $	an x = \frac{\sin x}{\cos x}$, and the derivative of $\cos x$ is $-\sin x$ — which appears in the numerator. This suggests a $u$-substitution with $u = \cos x$, turning the integral into $\int \frac{-du}{u}$, a standard logarithmic form.

The limits $0$ to $\pi/4$ are chosen because $	an x$ is well-behaved there (no vertical asymptotes), and the result simplifies nicely to a neat logarithmic value.

## Step-by-Step Solution

1. **Rewrite the integrand**  
   Start by expressing $	an x$ in terms of sine and cosine:  
   $$ \int_0^{\pi/4} 	an x \, dx = \int_0^{\pi/4} \frac{\sin x}{\cos x} \, dx $$

2. **Choose a substitution**  
   Let $u = \cos x$. Then $du = -\sin x \, dx$, so $\sin x \, dx = -du$. This substitution works because the numerator $\sin x \, dx$ is exactly the differential of $\cos x$ (up to a sign).

3. **Change the limits of integration**  
   When $x = 0$, $u = \cos 0 = 1$.  
   When $x = \pi/4$, $u = \cos(\pi/4) = \frac{\sqrt{2}}{2}$.  
   The upper limit becomes smaller than the lower limit — this is fine; we'll handle it with the sign.

4. **Transform the integral**  
   Substitute everything:  
   $$ \int_{x=0}^{x=\pi/4} \frac{\sin x}{\cos x} \, dx = \int_{u=1}^{u=\sqrt{2}/2} \frac{1}{u} \cdot (-du) = -\int_{1}^{\sqrt{2}/2} \frac{du}{u} $$

5. **Reverse the limits to simplify**  
   The negative sign can be absorbed by swapping the limits:  
   $$ -\int_{1}^{\sqrt{2}/2} \frac{du}{u} = \int_{\sqrt{2}/2}^{1} \frac{du}{u} $$

6. **Integrate** …

Q.Evaluate the definite integral: $\int_0^{\pi/4} \tan x \ dx$

Concept understanding — Integrals of Trigonometric Functions (tan x, cot x, sec x, cosec x)

Why This Approach Works

Step-by-Step Solution

Method: Definite integral of $\tan x$ via a logarithm

Steps

Common Mistakes

Q.Evaluate the definite integral: ∫0π/4tan⁡x dx\int_0^{\pi/4} \tan x \ dx∫0π/4​tanx dx

Why This Approach Works

Step-by-Step Solution

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Q.Evaluate the definite integral: $\int_0^{\pi/4} \tan x \ dx$

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