Find the integral of the function cos4 2x - Vidyadip

Question

Find the integral of the function cos⁴ 2x

✓

Answer

cos⁴2x = (cos²2x)²
$= \left(\frac{1+\cos 4 x}{2}\right)^{2}$
$= \frac{1}{4}\left[1+\cos ^{2} 4 x-2 \cos 4 x\right]$
$= \frac{1}{4}\left[1+\left(\frac{1+\cos 8 x}{2}\right)+2 \cos 4 x\right]$
$= \frac{1}{4}\left[\frac{3}{2}+\frac{1}{2} \cos 8 x+2 \cos 4 x\right]$
Now, $\int \cos ^{4} 2 x d x=\int\left[\frac{3}{8}+\frac{1}{8} \cos 8 x+\frac{1}{2} \cos 4 x\right] d x$
$= \frac{3 x}{8}+\frac{1}{64} \sin 8 x+\frac{1}{8} \sin 4 x+C$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks" from Integrals→Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following integrals:

$\int\text{x}^3\cos\text{x}^2\text{dx}$

Solve the following linear programming problem graphically:
Maximise Z = 4x + y subject to the constraints:
x + y $\le$ 50
3x + y $\le$ 90
x $\ge$ 0, y $\ge$ 0

Evaluate the following integrals:
$=\int\frac{\sin\text{x}}{(1+\cos\text{x})^2}\text{dx}$

The side of a square sheet is increasing at the rate of 4cm per minute. At what rate is the area increasing when the side is 8cm long?

Show that the following systems of linear equations has infinite number of solutions and solve:

x - y + 3z = 6,

x + 3y - 3z = -4,

5x + 3y + 3z = 10

Evaluate $\sin\Big(\tan^{-1}\frac{3}{4}\Big).$

The items produced by a company contain 10% defective items. Show that the probability of getting 2 defective items in a sample of 8 items is $\frac{28\times9^6}{10^8}$.

Let $\text{A} = \begin{bmatrix}0 & 1\\0 & 0 \end{bmatrix},$show that (aI + bA)ⁿ = aⁿI + na^n-1 bA where I is the identity matrix of order 2 and $ \text{n} \in \text{N}.$

The total cost of producing x radio sets per day is Rs $\Big(\frac{\text{x}^{2}}{4}+35\text{x}+25\Big)$ and the price per set at which they may be sold is Rs $(50-\frac{\text{x}}{2})$. Find the daliy output to maximum the tatal profit.

In the following, determine the values of constants involved in the definition so that the given function is continuous:
$\text{f(x)}=\begin{cases}\text{kx}+5,&\text{if }\text{ x}\leq2\\\text{x}-1,&\text{if }\text{ x}>2\end{cases}$