Integral Calculus-1
Integral Calculus-1
Allen Star Batch
Grade 12
Question:
If $\int \sqrt{\frac{\cos x - \cos^3 x}{\left(1-\cos^3 x\right)}} dx = f(x) + c$, then $f(x)$ is equal to:
$\frac{2}{3}\sin^{-1}\left(\cos^3 x\right)$
$\frac{3}{2}\sin^{-1}\left(\cos^3 x\right)$
$\frac{2}{3}\cos^{-1}\left(\cos^3 x\right)$
$-\frac{2}{3}\sin^{-1}\left(\cos^3 x\right)$
Step-by-Step Solution
Key Concept: Recognize that the integrand simplifies with the substitution $\cos^2 x = p$, reducing it to a standard inverse trigonometric integral.
For $I = \int \sqrt{\frac{\cos x - \cos^3 x}{1-\cos^3 x}}dx$, substitute $\cos^2 x = p$ so that $-\frac{3}{2}\cos^2 x\sin x\,dx = dp$. This transforms the integral to $I = -\frac{2}{3}\int \frac{dp}{\sqrt{1-p^2}} = -\frac{2}{3}\sin^{-1}(\cos^2 x) + C = \frac{2}{3}\cos^{-1}(\cos^2 x) + C_1$.
Correct Answer: 3,4