Q.Integrate the following function: $\sin^4 x$

Concept: Sine Power Reduction — use the double-angle identity $\sin^2 x = \frac{1 - \cos 2x}{2}$ repeatedly to lower the power.

Step 1: Write $\sin^4 x = (\sin^2 x)^2 = \left(\frac{1 - \cos 2x}{2}\right)^2 = \frac{1}{4}(1 - 2\cos 2x + \cos^2 2x)$.

Step 2: Reduce $\cos^2 2x$ using $\cos^2 \theta = \frac{1 + \cos 2\theta}{2}$:

$\cos^2 2x = \frac{1 + \cos 4x}{2}$.

So $\sin^4 x = \frac{1}{4}\left(1 - 2\cos 2x + \frac{1 + \cos 4x}{2}\right) = \frac{1}{4}\left(\frac{3}{2} - 2\cos 2x + \frac{1}{2}\cos 4x\right)$. …

Why power reduction works

Integrating $\sin^4 x$ directly is messy — you’d need to expand $(\sin^2 x)^2$ and then use identities, but that’s error-prone. The cleanest path is to use the sine power-reduction formula, which comes from the double-angle identity for cosine:

This formula lets you replace a square of sine with a linear expression in cosine. Applying it twice — once to $\sin^2 x$, then again to the resulting $\sin^2 2x$ — reduces the fourth power to a sum of cosines that are trivial to integrate.

Step-by-step integration

1. Rewrite $\sin^4 x$ as $(\sin^2 x)^2$ and apply the formula once.

2. Now handle $\cos^2 2x$ using the same idea.

The double-angle identity for cosine also gives a power-reduction formula for cosine:

Here $\theta = 2x$, so $\cos^2 2x = \frac{1 + \cos 4x}{2}$.

3. Substitute back into the expression.

4. Simplify the constant term and the coefficients.

First, combine the constants inside the parentheses:

So

Multiply through by $\frac{1}{4}$:

5. Integrate term by term.

Each term is straightforward:

• $\int \frac{3}{8} \, dx = \frac{3}{8}x$. …

Method: Repeated power reduction for $\sin^4 x$ (and higher even powers)

An even power like $\sin^4 x$ is reduced by applying the double-angle identity, then reducing the leftover squared cosine a second time.

Steps

Step 1: Reduce the outer square.

Step 2: Reduce the leftover $\cos^2 2x$.

Substitute this in so only first-degree cosines remain.

Step 3: Integrate term by term. …

Common Mistakes

Mistake 1: Reducing $\sin^4 x$ only once and integrating $\cos^2 2x$ directly.

Why it's wrong: after $\sin^4 x=\tfrac14(1-2\cos 2x+\cos^2 2x)$, the $\cos^2 2x$ is still an even power and cannot be integrated as-is. Correct approach: reduce again with $\cos^2 2x=\tfrac{1+\cos 4x}{2}$.

Mistake 2: Using a single $\dfrac1k$ divisor for both $\cos 2x$ and $\cos 4x$. …