Find the integral of the function ^4 2x - Vidyadip

Question

Find the integral of the function $\cos^4 2x$

✓

Answer

$\cos^42x = (\cos^22x)^2$
$= \left(\frac{1+\cos 4 x}{2}\right)^{2}$
$= \frac{1}{4}\left[1+\cos ^{2} 4 x-2 \cos 4 x\right]$
$= \frac{1}{4}\left[1+\left(\frac{1+\cos 8 x}{2}\right)+2 \cos 4 x\right]$
$= \frac{1}{4}\left[\frac{3}{2}+\frac{1}{2} \cos 8 x+2 \cos 4 x\right]$
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 Question" 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

Solve the following equation for x:
$\tan^{-1}\frac{\text{x}-2}{\text{x}-1}+\tan^{-1}\frac{\text{x}+2}{\text{x}+1}=\frac{\pi}{4}$

Prove the following results:
$4\tan^{-1}\frac{1}{5}-\tan^{-1}\frac{1}{239}=\frac{\pi}{4}$

Differentiate the following functions with respect to x:
$\tan(\text{e}^{\sin\text{x}})$

Evaluate the following definite integrals:
$\int_{1}^\limits{3}\frac{\log\text{x}}{(\text{x}+1)^2}\text{ dx}$

On the set Q of all ration numbers if a binary operation * is defined by $\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{5},$ prove that * is associative on Q.

If $\text{A}=\begin{bmatrix}2&3\\5&7\end{bmatrix},\text{ B}=\begin{bmatrix}-1&0&2\\3&4&1\end{bmatrix},\text{C}=\begin{bmatrix}-1&2&3\\2&1&0\end{bmatrix},$ find2B + 3A and 3C - 4B.

Let $+_6 ($addition modulo $6) $ be a binary operation on $S = \{0, 1, 2, 3, 4, 5\}$. Write the value of $2 + _64^{−1}+ _63^{−1}.$

Write the anti-derivative of $\Big(3\sqrt{\text{x}}+\frac{1}{\sqrt{\text{x}}}\Big)$

For the binary operation multiplication modulo $10 (\times _{10})$ defined on the set $S = \{1, 3, 7, 9\},$ write the inverse of $3.$

The probability of a man hitting a target is 1/4. If he fires 7 times, what is the probability of his hitting the target at least twice?