Q.Integrate the following function: $\sin 2x$

Step 0 — The limits.

The printed limits render as $\pm\tfrac12$, but with those limits the value would be $\sin(0.5)\approx 0.479$, which is not among the options. The intended (and standard KCET) integral is over the symmetric interval $\left[-\tfrac{\pi}{2},\tfrac{\pi}{2}\right]$ — the $\pi$ has been lost in typesetting. Every option is consistent with that reading.

Step 1 — Set up the king's-property trick.

Replace $x\to -x$ (limits are symmetric, so the value is unchanged):

using $\cos(-x)=\cos x$ (even) and $\dfrac{1}{1+e^{-x}}=\dfrac{e^{x}}{1+e^{x}}$.

Step 2 — Add the two expressions.

Step 3 — Evaluate.

$\cos x$ is even, so

ANSWER: C

Step 1 — Turn the radicand into a perfect square.

For any angle $\theta$,

This is the standard trick for a $\sqrt{1\pm\sin\theta}$ integral: a square root is only integrable in closed form once the radicand is a square.

Step 2 — Apply it with $\theta=\dfrac{x}{2}$ (so $\dfrac{\theta}{2}=\dfrac{x}{4}$).

Step 3 — Remove the modulus (this is where such problems usually go wrong).

As $x$ runs over $[0,2\pi]$, the argument $\dfrac{x}{4}$ runs over $\left[0,\dfrac{\pi}{2}\right]$. In the first quadrant both $\sin\frac{x}{4}\ge 0$ and $\cos\frac{x}{4}\ge 0$, so the bracket never changes sign and

No splitting of the interval is needed.

Step 4 — Integrate.

(Each antiderivative carries a factor $\dfrac{1}{1/4}=4$.)

Step 5 — Evaluate.

At $x=2\pi$: $\dfrac{x}{4}=\dfrac{\pi}{2}$, giving $-4\cos\dfrac{\pi}{2}+4\sin\dfrac{\pi}{2}=0+4=4$.

At $x=0$: $-4\cos 0+4\sin 0=-4$.

ANSWER: A

Step 1 — The symmetry property to use

For an interval symmetric about the origin,

Here $a = \pi$, so the parity of the integrand settles everything.

Step 2 — Determine the parity of each factor

• $1 - x^2$: replacing $x \to -x$ gives $1 - x^2$ ⇒ even
• $\sin x$: $\sin(-x) = -\sin x$ ⇒ odd
• $\cos^2 x$: $\cos^2(-x) = (\cos x)^2 = \cos^2 x$ ⇒ even

Step 3 — Combine

So $f$ is an odd function (even $\times$ even $\times$ odd $=$ odd).

Step 4 — Apply the property

Why this is intuitive: for every contribution $f(x)\,dx$ on the right of the origin there is an exactly equal and opposite contribution $f(-x)\,dx$ on the left; the signed areas cancel in pairs.

ANSWER: D

Concept: simplify the trigonometric integrand before integrating.

Denominator: cot x - tan x = cos x/sin x - sin x/cos x = (cos^2 x - sin^2 x)/(sin x cos x) = cos 2x / ((1/2) sin 2x) = 2 cos 2x / sin 2x = 2 cot 2x.

Numerator: cos 4x + 1 = 2 cos^2 2x.

So the integrand becomes

(2 cos^2 2x) / (2 cot 2x) = cos^2 2x * tan 2x = cos^2 2x * (sin 2x / cos 2x) = sin 2x cos 2x = (1/2) sin 4x.

Integrate:

integral (1/2) sin 4x dx = (1/2) * ( -cos 4x / 4 ) + c = -(1/8) cos 4x + c.

ANSWER: B

Step 1 — The key identity. From $\cos\phi=2\cos^2\frac{\phi}{2}-1$,

Step 2 — Innermost radical. The stem's innermost term is $2+2\cos 8\theta$ (the three nested radicals require the inner angle $8\theta$; with a bare $\theta$ the expression would reduce to $2\cos\frac{\theta}{8}$, which is not among the options):

Step 3 — Second radical.

Step 4 — Outermost radical.

Step 5 — Check with a value. Take $\theta=\dfrac{\pi}{8}$, so $8\theta=\pi$ and $\cos 8\theta=-1$: innermost $=\sqrt{2-2}=0$, then $\sqrt{2+0}=\sqrt2$, then $\sqrt{2+\sqrt2}\approx1.8478$. And $2\cos\frac{\pi}{8}=2(0.92388)=1.8478$ ✓.

ANSWER: A

Step 1 — Convert to degrees to see the structure.

Notice at once that $75^\circ + 15^\circ = 90^\circ$ — the angles are complementary. That is the hook the question is built on.

Step 2 — Use the complementary (co-function) identity.

So the required product becomes a sine × cosine of the same angle:

Step 3 — Apply the double-angle identity.

Since $\sin 2\theta = 2\sin\theta\cos\theta$, we have $\sin\theta\cos\theta = \tfrac12 \sin 2\theta$. With $\theta = 75^\circ$:

Step 4 — Evaluate.

Step 5 — Independent verification (product-to-sum).

Use $2\sin A\sin B = \cos(A-B) - \cos(A+B)$ with $A = 75^\circ,\ B = 15^\circ$:

A numerical check also agrees: $\sin 75^\circ \approx 0.9659$, $\sin 15^\circ \approx 0.2588$, and $0.9659 \times 0.2588 \approx 0.250$. ✓

ANSWER: C