The value of ^ _01 5+3 dx is : - Vidyadip

MCQ

The value of $\int\limits^{\pi}_0\frac{1}{5+3\cos\text{x}}\text{ dx}$ is :

✓
$\frac{\pi}{4}$
B
$\frac{\pi}{8}$
C
$\frac{\pi}{2}$
D
$0$

✓

Answer

Correct option: A.

$\frac{\pi}{4}$

$\int\limits^{\pi}_0\frac{1}{5+3\cos\text{x}}\text{ dx}$
$=\int\limits^{\pi}_0\frac{1}{5+3\frac{1-\tan^{2}\frac{\text{x}}{2}}{1+\tan^{2}\frac{\text{x}}{2}}}\text{ dx}$
$=\int\limits^{\pi}_0\frac{1+\tan^{2}\frac{\text{x}}{2}}{5+5\tan^{2}\frac{\text{x}}{2}+3-3\tan^{2}\frac{\text{x}}{2}}$
$=\int\limits^{\pi}_0\frac{\sec^2\frac{\text{x}}{2}}{8+2\tan^{2}\frac{\text{x}}{2}}\text{ dx}$
Let $\tan\frac{\text{x}}{2}=\text{t},$ then $\sec^2\frac{\text{x}}{2}\text{ dx}=2\text{dt}$
When $\text{x}=0,\text{ t}=0,\text{x}=\pi,\text{ t}=\infty$
Therefore the integral becomes
$\frac{1}{2}\int\limits^{\infty}_0\frac{\text{dt}}{4+\text{t}^2}$
$=\frac{1}{2}\Big[\tan^{-1}\frac{\text{t}}{2}\Big]^{\infty}_0$
$=\frac{1}{2}\Big(\frac{\pi}{2}-0\Big)$
$=\frac{\pi}{4}$

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 "M.C.Q (1 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

Let $\vec{a}=2 \hat{i}+7 \hat{j}-\hat{k}, \vec{b}=3 \hat{i}+5 \hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+2 \hat{k}$ Let $\vec{d}$ be a vector which is perpendicular to both $\overrightarrow{ a }$ and $\overrightarrow{ b }, \quad$ and $\quad \overrightarrow{ c } \cdot \overrightarrow{ d }=12$. Then $(-\hat{i}+\hat{j}-\hat{k}) \cdot(\vec{c} \times \vec{d})$ is equal to $........$.

The number of real roots of the equation $e^{6 x}-e^{4 x}-2 e^{3 x}-12 e^{2 x}+e^{x}+1=0$ is:

If $\int\text{x}\sin\text{x dx}=-\text{x}\cos\text{x}+\text{a},$ then a is equal to :

If $\vec{\text{a}},\ \vec{\text{b}}$ are the vectors forming consecutive sides of a regular hexagon ABCDEF, then the vector representing side CD is,

The solution set of the inequation $3x + 2y > 3$ is:

If $\text{y}^\frac{1}{\text{n}}+\text{y}-^\frac{1}{\text{n}}=2\text{x}$ then find $(\text{x}^2-1)\text{y}_2+\text{xy}_1=$

A binary operation $*$ on $Z$ defined by $a^ * b = 3a + b$ for all $a, b \in Z,$ is:

The area bounded by the curve $y = x|x|$ and the ordinates $x = -1$ and $x = 1$ is given by:

If $y(x)=x^2, x > 0$, then $y^{\prime \prime}(2)-2 y^{\prime}(2)$ is equal to :

Let $v$ be the solution of the differential equation $\left(1-x^{2}\right) d y=\left(xy+\left(x^{3}+2\right) \sqrt{1-x^{2}}\right) d x,-1 < x < 1$ and $y(0)=0$ if $\int\limits_{-\frac{1}{2}}^{\frac{1}{2}} \sqrt{1-x^{2}} y(x) d x=k$ then $k^{-1}$ is equal to: