Evaluate the following integrals: 8x 9+ ^44x dx - Vidyadip

Question

Evaluate the following integrals:$\int\frac{\sin8\text{x}}{\sqrt{9+\sin^44\text{x}}}\text{ dx}$

✓

Answer

Let $\text{I}=\int\frac{\sin8\text{x}}{\sqrt{9+(\sin4\text{x})^4}}\text{ dx}$
Let $\sin^24\text{x}=\text{t}$
$\Rightarrow2\sin4\text{x}.\cos4\text{x}(4)\text{dx}=\text{dt}$
$\Rightarrow4\sin8\text{x}\text{ dx}=\text{dt}$
$\Rightarrow\sin8\text{x}\text{ dx}=\frac{\text{dt}}{4}$
$\text{I}=\frac{1}{4}\int\frac{\text{dt}}{\sqrt{(3)^2+\text{t}^2}}$
$\text{I}=\frac{1}{4}\log\Big|\text{t}+\sqrt{(3)^2+\text{t}^2}\Big|+\text{C}$ $\Big[\text{Since }\int\frac{1}{\sqrt{\text{a}^2+\text{x}^2}}\text{ dx}=\log\Big|\text{x}+\sqrt{\text{a}^2+\text{x}^2}\Big|+\text{C}\Big]$
$\text{I}=\frac{1}{4}\log\Big|\sin^24\text{x}+\sqrt{9+\sin^44\text{x}}\Big|+\text{C}$

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 "Solve the Following Question.(3 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

If $|\vec{\text{a}}|=\text{a}$ and $\big|\vec{\text{b}}\big|=\text{b},$ prove that $\Big(\frac{\vec{\text{a}}}{\text{a}^2}-\frac{\vec{\text{b}}}{\text{b}^2}\Big)^2=\Big(\frac{\vec{\text{a}}-\vec{\text{b}}}{\text{ab}}\Big)^2.$

Evaluate the following integrals:$\int_{0}^\limits{1}\frac{2\text{x}}{1+\text{x}^4}\text{ dx}$

The binary operation * is defined by $\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{7}$ on the set Q of all rational numbers. Show that * is associative.

Differentiate the following functions with respect to x:
$\tan^2\text{x}$

If $\int\text{e}^{\text{x}}(\tan\text{x}+1)\sec\text{x dx}=\text{e}^{\text{x}}\text{ f}(\text{x})+\text{C},$ then write the value of f(x).

Evaluate the following integrals:
$\int\frac{\big\{\text{e}^{\sin^{-1}\text{x}}\big\}^2}{\sqrt{1-\text{x}^2}}\text{dx}$

Show that the following planes are at right angles.
$\vec{\text{r}}\cdot(2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}})=5$ and $\vec{\text{r}}\cdot(-\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}})=3$

Given $\text{A}=\begin{bmatrix}2 & -3 \\-4 & 7 \end{bmatrix},$ compute $A^{-1}$ and show that $2A^{-1} = 9I - A$.

Find the angle between the given planes.
$\vec{\text{r}}\cdot(2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}})=1$ and $\vec{\text{r}}\cdot(-\hat{\text{i}}+\hat{\text{j}})=4$

Solve the following differential equations:

$\frac{d y}{d x}+y \cdot \sec x=\tan x$