Find dθ d ( sin³ θ ) (i) Use the substitution x = tan θ to evaluate ∫₀¹ (x²)² (1 + x²)² dx - HSC - SSCE Mathematics Extension 1 - Question 13 - 2020 - Paper 1

Using the substitution $x = \tan \theta$, we have:

$dx = \sec^2 \theta \, d\theta$

The limits change as follows:

• When $x=0$, $\theta=0$;
• When $x=1$, $\theta=\frac{\pi}{4}$.

Now we can rewrite the integral:

$\int_0^1 \frac{x^4}{(1+x^2)^2} \, dx = \int_0^{\frac{\pi}{4}} \frac{\tan^4 \theta}{(1+\tan^2 \theta)^2} \sec^2 \theta \, d\theta$

Since $1 + \tan^2 \theta = \sec^2 \theta$, this can simplify to:

$\int_0^{\frac{\pi}{4}} \tan^4 \theta \, d\theta$.

We can further simplify and evaluate this integral using known trigonometric identities.

To find the volume of the solid created by rotating region R about the x-axis, we can use the disk method. The volume V can be calculated as:

$V = \pi \int_a^b [f(x)]^2 \, dx$

In this case, the boundaries of the region can be determined from $y = sin x$ and $y = cos(2x)$.

We first find the intersection points by solving:

$cos(2x) = sin x$.

Solving gives us the limits of integration. Then:

$V = \pi \int_a^b [cos(2x)]^2 \, dx - \pi \int_a^b [sin x]^2 \, dx$.

Calculating these integrals will yield the final volume.