Evaluate the integral: I = integral cos^5x sqrt(sin x) dx
Evaluate the integral: $$ I = \int \cos^5x\,\sqrt{\sin x}\,dx. $$
Explanation of Method
Since the power of cosine is odd, peel off one factor of $\cos x$ to pair with $dx$, rewrite the remaining even power $\cos^4x=(1-\sin^2x)^2$, and substitute $u=\sin x$. The whole integrand becomes a polynomial in fractional powers of $u$.
- Separate one factor of $\cos x$ from the odd power.
- Substitute $u=\sin x$ and expand.
- Integrate term by term.
- Substitute back and add the constant of integration.
Step 1 · Separate one cosine factor
We separate one cosine factor and rewrite the rest via the Pythagorean identity:
$$ \cos^5x = \cos^4x\cdot\cos x = (1-\sin^2x)^2\cos x. $$
Thus, the integral becomes:
$$ I = \int (1-\sin^2x)^2\,\sqrt{\sin x}\,\cos x\,dx. $$
Step 2 · Substitution
Let:
$$ u = \sin x \quad \text{so that} \quad du = \cos x\,dx. $$
Substituting into the integral:
$$ I = \int (1-u^2)^2\,u^{1/2}\,du. $$
Step 3 · Expand and simplify
Expanding $(1-u^2)^2=1-2u^2+u^4$ and distributing $u^{1/2}$:
$$ I = \int \left(u^{1/2} - 2u^{5/2} + u^{9/2}\right) du. $$
Step 4 · Integrate term by term
Integrating each term using the power rule:
$$ I = \frac{2}{3}u^{3/2} - \frac{4}{7}u^{7/2} + \frac{2}{11}u^{11/2} + C. $$
Step 5 · Substitute back
Substituting back $u=\sin x$:
$$ I = \frac{2}{3}\sin^{3/2}x - \frac{4}{7}\sin^{7/2}x + \frac{2}{11}\sin^{11/2}x + C. $$
Summary Table
| Term | Antiderivative |
|---|---|
| $u^{1/2}$ | $\tfrac23u^{3/2}$ |
| $-2u^{5/2}$ | $-\tfrac47u^{7/2}$ |
| $u^{9/2}$ | $\tfrac{2}{11}u^{11/2}$ |
Additional Notes
- The integrand requires $\sin x \geq 0$ for the square root to be real, restricting $x$ to intervals such as $[0,\pi]$.
- The odd-power-splitting trick works just as well when the remaining substitution variable appears to a fractional power, as here — the exponents simply shift by $\tfrac12$ throughout.
- The result can be checked by differentiating $\tfrac23\sin^{3/2}x-\tfrac47\sin^{7/2}x+\tfrac{2}{11}\sin^{11/2}x$ with the chain rule and confirming it returns $\cos^5x\sqrt{\sin x}$.
Result
The solution to the integral is: