Evaluate the definite integral:

$\int_{0}^{2/3} \frac{dx}{4+9x^2}$ equals (A) $\frac{\pi}{6}$ (B) $\frac{\pi}{12}$ (C) $\frac{\pi}{24}$ (D) $\frac{\pi}{4}$

Question

Evaluate the definite integral:

$\int_{0}^{2/3} \frac{dx}{4+9x^2}$ equals (A) $\frac{\pi}{6}$ (B) $\frac{\pi}{12}$ (C) $\frac{\pi}{24}$ (D) $\frac{\pi}{4}$

Accepted Answer

> [!TLDR]
> The integral $\int_{0}^{2/3} \frac{dx}{4+9x^2}$ is solved by recognizing the form $\frac{1}{a^2 + u^2}$ and using the substitution $u = 3x$, leading to the result $\frac{\pi}{24}$, which corresponds to option **(C)**.

The key to this problem is spotting that the denominator $4 + 9x^2$ looks like $a^2 + u^2$, the classic form for an inverse tangent integral. The standard formula is:

> [!FORMULA]
> $$ \int \frac{du}{a^2 + u^2} = \frac{1}{a} 	an^{-1}\left(\frac{u}{a}ight) + C $$

Here, $4$ is $2^2$, and $9x^2$ is $(3x)^2$. So we have $a = 2$ and $u = 3x$. The substitution $u = 3x$ will cleanly match the formula.

Let’s work through it step by step.

1. **Set up the substitution.**  
   Let $u = 3x$. Then $du = 3\,dx$, so $dx = \frac{du}{3}$.  
   The limits change: when $x = 0$, $u = 0$; when $x = \frac{2}{3}$, $u = 3 \cdot \frac{2}{3} = 2$.

2. **Rewrite the integral.**  
   Substitute everything into the integral:
   $$ \int_{0}^{2/3} \frac{dx}{4 + 9x^2} = \int_{u=0}^{u=2} \frac{du/3}{4 + u^2} = \frac{1}{3} \int_{0}^{2} \frac{du}{2^2 + u^2} $$

3. **Apply the inverse tangent formula.**  
   Using $\int \frac{du}{a^2 + u^2} = \frac{1}{a} 	an^{-1}\left(\frac{u}{a}ight)$ with $a = 2$:
   $$ \frac{1}{3} \left[ \frac{1}{2} 	an^{-1}\left(\frac{u}{2}ight) ight]_{0}^{2} = \frac{1}{6} \left[ 	an^{-1}\left(\frac{u}{2}ight) ight]_{0}^{2} $$

4. **Evaluate the limits.**  
   At $u = 2$: $	an^{-1}\left(\frac{2}{2}ight) = 	an^{-1}(1) = \frac{\pi}{4}$.  
   At $u = 0$: $	an^{-1}(0) = 0$.  
   So the result is:
   $$ \frac{1}{6} \left( \frac{\pi}{4} - 0 ight) = \frac{\pi}{24} $$ …

Q.Evaluate the definite integral: $\int_{0}^{2/3} \frac{dx}{4+9x^2}$ equals (A) $\frac{\pi}{6}$ (B) $\frac{\pi}{12}$ (C) $\frac{\pi}{24}$ (D) $\frac{\pi}{4}$

Concept understanding — U Substitution

U Substitution: The Reverse Chain Rule

The Core Intuition

The Precise Statement

The Step-by-Step Method

A Second Example (with a constant factor)

When Does It Work?

Method: Reduce $\dfrac{1}{a^2+u^2}$ by a linear substitution to the arctan standard form

Steps

Common Mistakes

Concept and Intuition

Step-by-Step Solution

Concept and Intuition

Step-by-Step Solution

Common Mistakes …

Q.Evaluate the definite integral: ∫02/3dx4+9x2\int_{0}^{2/3} \frac{dx}{4+9x^2}∫02/3​4+9x2dx​ equals (A) π6\frac{\pi}{6}6π​ (B) π12\frac{\pi}{12}12π​ (C) π24\frac{\pi}{24}24π​ (D) π4\frac{\pi}{4}4π​

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Q.Evaluate the definite integral: $\int_{0}^{2/3} \frac{dx}{4+9x^2}$ equals (A) $\frac{\pi}{6}$ (B) $\frac{\pi}{12}$ (C) $\frac{\pi}{24}$ (D) $\frac{\pi}{4}$

Unlock everything free for 14 days