{"id":37995,"date":"2026-02-20T06:41:15","date_gmt":"2026-02-20T06:41:15","guid":{"rendered":"https:\/\/mp.moonpreneur.com\/math-corner\/?p=37995"},"modified":"2026-03-07T10:15:45","modified_gmt":"2026-03-07T10:15:45","slug":"wallis-formula-integration-sine-cosine","status":"publish","type":"post","link":"https:\/\/mp.moonpreneur.com\/math-corner\/wallis-formula-integration-sine-cosine\/","title":{"rendered":"The Wallis Formula: Integrating Powers of Sine and Cosine Instantly"},"content":{"rendered":"\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
\n\t\t\t\t\t
\n\t\t\t
\n\t\t\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\t\t\t
\n\t\t\t\t

If you are a student preparing for the SAT, AP Calculus, or competitive exams like the JEE, you know the feeling of dread when you see a trig function raised to a massive power inside an integral. Imagine seeing \\(\\displaystyle \\int_{0}^{\\pi\/2} \\sin^{-7}(x)\\,dx\\) <\/span>and thinking you have to use integration by parts or reduction formulas over and over again. It\u2019s exhausting!<\/span><\/p>

But what if I told you there\u2019s a “magic” formula that lets you solve these in seconds? It\u2019s called the <\/span>Wallis Formula<\/b>, named after the great mathematician John Wallis. This isn’t just a shortcut; it’s a game-changer for your math toolkit.<\/span><\/p>

The Golden Rule: Check Your Boundaries<\/b><\/span><\/h3>

Before we dive into the math, there is one non-negotiable rule. The Wallis Formula <\/span>only<\/b> works if your integration interval is from <\/span>0<\/span> to <\/b>\u03c0<\/span><\/i>\/2<\/span>. If you see those limits and a power of sine or cosine, you\u2019re ready to use the hack.<\/span><\/p>

Interestingly, the formula works exactly the same for both \\(\\displaystyle \\sin^{n}(x)\\) and \\(\\displaystyle \\cos^{n}(x)\\) .\\<\/span><\/p>

The Wallis Formula allows for the instant evaluation of definite integrals of \\(\\displaystyle \\sin^{n}(x)\\) or \\(\\displaystyle \\cos^{n}(x)\\) from 0 to \u03c0\/2 without standard integration techniques. For a non-negative integer n, the integral is\u00a0 \\(\\displaystyle \\frac{(n – 1)!!}{n!!}\\) \u22c5 k, where k = \u03c0\/2 if n is even and k = 1 if n is odd.<\/p>

The core Formulas<\/strong><\/span><\/h3>

For an integer n \u2265 0, the integral is defined as:\u00a0\u00a0<\/p>

\\(\\displaystyle I_n = \\int_{0}^{\\pi\/2} \\sin^{n}(x)\\,dx
= \\int_{0}^{\\pi\/2} \\cos^{n}(x)\\,dx\\)<\/p>

The result depends entirely on whether n is odd<\/strong> or even<\/strong>:<\/p>

Even power Case(n is even):<\/strong><\/span><\/p>

\\(\\displaystyle
\\int_{0}^{\\pi\/2} \\sin^{n}(x)\\,dx
= \\frac{n-1}{n} \\cdot \\frac{n-3}{n-2} \\cdots \\frac{1}{2} \\cdot \\frac{\\pi}{2}
\\quad \\text{(for even } n\\text{)}
\\)<\/p>

Odd Power Case (n is odd):<\/strong>\u00a0<\/span><\/p>

\\(\\displaystyle
\\int_{0}^{\\pi\/2} \\sin^{n}(x)\\,dx
= \\frac{n-1}{n} \\cdot \\frac{n-3}{n-2} \\cdots \\frac{2}{3} \\cdot 1
\\quad \\text{(for odd } n\\text{)}
\\)<\/p>

How It Works: Even vs. Odd Powers<\/b><\/span><\/h3>

The beauty of this formula is its symmetry. You just need to look at the exponent (<\/span>n<\/span><\/i>) and decide if it is <\/span>even<\/b> or <\/span>odd.<\/b><\/p>

Case 1: The Exponent is Odd (e.g., n<\/i>=3,5,7,9…)<\/strong><\/span><\/h5>

The pattern is simple: start with <\/span>(<\/span>n<\/span><\/i>\u22121)<\/span> in the numerator and <\/span>n<\/span><\/i> in the denominator. Then, keep subtracting 2 from both until you hit the end of the line.<\/span><\/p>

The Pattern: \\(\\displaystyle
\\frac{n-1}{n} \\times \\frac{n-3}{n-2} \\times \\frac{n-5}{n-4}
\\) ending at \\(\\displaystyle \\frac{2}{3} \\times 1\\)<\/p>

Example: Let’s Solve \\(\\displaystyle \\int_{0}^{\\pi\/2} \\sin^{7}(x)\\,dx\\)<\/strong><\/p>

  1. Start with <\/span>n <\/span><\/i>= 7<\/span>.<\/span><\/li>
  2. Numerator: <\/span>7 \u2212 1 = 6 and Denominator: <\/span>7.<\/span>\u00a0Fraction = <\/span>6\/7<\/span>.<\/span><\/li>
  3. Subtract 2: <\/span>6 \u2212 2 = 4<\/span>. <\/span>7 \u2212 2 = 5<\/span>. Fraction = <\/span>4\/5<\/span>.<\/span><\/li>
  4. Subtract 2 again: <\/span>4 \u2212 2 = 2<\/span>.\u00a0 <\/span>5 \u2212 2 = 3<\/span>. Fraction = <\/span>2\/3<\/span>.<\/span><\/li>
  5. Multiply them: <\/span>(6\/7) \u00d7 (4\/5) \u00d7 (2\/3) = <\/span>16\/35<\/b><\/li><\/ol>
    Case 2: The Exponent is Even (e.g., <\/b>n<\/i><\/b>=2,4,6…)<\/b><\/span><\/h5>

    The pattern is almost the same, but with one “special ingredient” at the end: multiply by <\/span>\u03c0<\/span><\/i>\/2.<\/span><\/p>