Evaluate the following integrals: _ 0 ^ 3 3+4 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits_{0}^{\frac{\pi}{3}}\frac{\cos\text{x}}{3+4\sin\text{x}}\text{ dx}$

✓

Answer

Let $\text{I}=\int_{0}^\limits{\frac{\pi}{3}}\frac{\cos\text{x}}{3+4\sin\text{x}}\text{ dx}$
Let $\sin\text{x}=\text{t}$ Then, $\cos\text{x}\text{dx}=\text{dt}$
When $\text{x}=0,\text{t}=0$ and $\text{x}=\frac{\pi}{3},\text{t}=\frac{\sqrt{3}}{2}$
$\therefore\ \text{I}=\int_{0}^\limits{\frac{\pi}{3}}\frac{\cos\text{x}}{3+4\sin\text{x}}\text{ dx}$
$=\int_{0}^\limits{\frac{\sqrt{3}}{2}}\frac{1}{3+4\text{t}}\text{ dt}$
$=\frac{1}{4}\big[\log\big(3+-4\text{t}\big)\big]^{\frac{\sqrt{3}}{2}}_0$
$=\frac{1}{4}\big(\log\big(3+2\sqrt{3}\big)-\log3\big)$
$=\frac{1}{4}\log\Big(\frac{3+2\sqrt{3}}{3}\Big)$

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 "4 Marks" from Definite Integrals→Definite 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 determinant equations:
$\begin{vmatrix}\text{x}+\text{a}&\text{b}&\text{c}\\\text{a}&\text{x}+\text{b}&\text{c}\\\text{a}&\text{b}&\text{x}+\text{c}\end{vmatrix}=0$

Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes $\vec{\text{r}}\cdot(\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}})=5$ and $\vec{\text{r}}\cdot(3\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})=6.$

Find the angle between two curves $y^2=4 a x$ and $x^2=4 b y$.

The equation of tangent at (2, 3) on the curve $y^{2} = \text{ax}^{3} + \text{b is y = 4x - 5}. $ Find the value of a and b.

Evaluate the following definite integrals:
$\int\limits_{0}^{1}\Big(\text{xe}^{\text{x}}+\cos\frac{\pi\text{x}}{4}\Big)\text{dx}$

Differentiate the following functions with respect to x:
$\text{e}^\text{x}\log\sin2\text{x}$

If $\text{y}=\tan^{-1}\Big(\frac{\sqrt{1+\text{x}}-\sqrt{1-\text{x}}}{\sqrt{1+\text{x}}+\sqrt{1+\text{x}}}\Big),$ find $\frac{\text{dy}}{\text{dx}}.$

Find the area of the region enclosed between the two curve x² + y² = 9 and (x - 3)² + y² = 9.

Solve the following differential equations:
$\frac{\text{dy}}{\text{dx}}=2\text{e}^{\text{x}}\text{y}^3,\text{y}(0)=\frac{1}{2}$

A dice rolled two times and sum of appeared number found 7 . Find the conditional probability of getting 3 at least one time.