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 "3 Marks Question" from Definite Integrals→Definite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

There are $6\%$ defective items in a large bulk of items. Find the probability that a sample of $8$ items will include not more than one defective item.

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

Find gof and fog when $f : R \rightarrow R$ and $g : R \rightarrow R$ are defined by:
$f(x) = x^2 + 2x - 3$ and $g(x) = 3x - 4$

Using vectors, find the value of $\lambda$ such that the points ($\lambda$, -10, 3), (1, -1, 3) and (3, 5, 3) are collinear.

$\text{If} \ \vec{a}=2\hat{i}+2\hat{j}+3\hat{k},\ \ \vec{b}=-$ $\hat{i}+2\hat{j}+\hat{k}\ \text{and}\ \vec{c}=3\hat{i}+\hat{j}$ are such that $\vec{a}+\lambda\vec{b}$ is perpendicular to $\vec{c},$ then find the value of $\lambda.$

Let $\text{A}=\begin{bmatrix}2&-3\\-7&5\end{bmatrix}$ and $\text{B}=\begin{bmatrix}1&0\\2&-4\end{bmatrix},$ verify that
$(\text{A}+\text{B})^\text{T}=\text{A}^\text{T}+\text{B}^\text{T}$

Evaluate the following:
$\int\limits^1_0\frac{\text{x}}{\sqrt{1+\text{x}^2}}\text{dx}$

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.$\text{f}(\text{x})=\sin\text{x}-\sin2\text{x}-\text{x}\text{ on }[0,\pi]$

Find the area of the pallallelogram determined by the vectors:$2\hat{\text{i}}$ and $3\hat{\text{j}}$

Discuss the applicability of the Rolle's theorem for the following function on the indicated interval
$\text{f}(\text{x})=\begin{cases}-4\text{x}+5,&0\leq\text{x}\leq1\\2\text{x}-3,&1<\text{x}\leq2\end{cases}$