Evaluate the following integrals: ^ _ 0 5 (5-4 )^ 1 4 d - Vidyadip

Question

Evaluate the following integrals:$\int^\limits{\pi}_{0}5\big(5-4\cos\theta\big)^{\frac{1}{4}}\sin\theta\text{ d}\theta$

✓

Answer

Let $\text{I}=\int^\limits{\pi}_{0}5\big(5-4\cos\theta\big)^{\frac{1}{4}}\sin\theta\text{ d}\theta$ Let $\big(5-4\cos\theta\big)=\text{t}$ Then, $4\sin\theta\text{ d}\theta=\text{dt}$ When $\theta=0,\text{t}=1$ and $\theta=\pi,\text{t}=9$$\therefore\ \text{I}=\frac{5}{4}\int\limits^9_1\text{t}^{\frac{1}{4}}\text{ dt}$
$\Rightarrow\text{I}=\frac{5}{4}\Bigg[\frac{4\text{t}^{\frac{5}{4}}}{5}\Bigg]^9_1$
$\Rightarrow\text{I}=\big(9\sqrt{3}-1\big)$

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 Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Differential equation $\frac{\text{dy}}{\text{dx}}+\text{y}=0,\text{y}(0)=3$Function $\text{y}=\text{e}^\text{-x}+2$

Diffrentiate the following w.r.t.x

$\log \left[4^{2 x}\left(\frac{x^2+5}{\sqrt{2 x^3-4}}\right)^{\frac{3}{2}}\right]$

The vector $\bar{a}$ is directed due north and $|\bar{a}|=24$. The vector $\bar{b}$ is directed due west and $|\bar{b}|$

$=7$. Find $|\bar{a}+\bar{b}|$.

Find the probability distribution of the number of doublets in 4 throws of a pair of dice.
Also, Find the mean and variance of this distribution.

If $\big[3\vec{\text{a}}+7\vec{\text{b}}\vec{\text{c}}\vec{\text{d}}\big]=\lambda\big[\vec{\text{a}}\vec{\text{c}}\vec{\text{d}}\big]+\mu\big[\vec{\text{b}}\vec{\text{c}}\vec{\text{d}}\big],$ then find the value of $\lambda+\mu.$

Find $f^{-1}$ if it exists: $f : A \rightarrow B$,where, $A = {0, -1, -3, 2}; B = {-9, -3, 0, 6}$ and $f(x) = 3x.$

Give an example of a function which is continuos but not differentiable at at a point.

Evaluate: $\int \frac{x^2 \cdot \tan ^{-1}\left(x^3\right)}{1+x^6} d x$

Check the commutativity and associativity of the following binary operations:
$'*'$ on Q defined by a * $b =( a - b )^2$ for $all a , b \in Q$.

Evaluate the following definite integrals:$\int_{1}^\limits{2}\text{e}^{2\text{x}}\Big(\frac{1}{\text{x}}-\frac{1}{2\text{x}^2}\big)\text{dx}$