Evaluvate the following intregals: 5 +6 2 + +3 dx - Vidyadip

Question

Evaluvate the following intregals:
$\int\frac{5\cos\text{x}+6}{2\cos\text{x}+\sin\text{x}+3}\ \text{dx}$

✓

Answer

Let $\text{I}=\int\frac{5\cos\text{x}+6}{2\cos\text{x}+\sin\text{x}+3}\ \text{dx}$
Let $(5\cos\text{x}+6)=\lambda\frac{\text{d}}{\text{dx}}(2\cos\text{x}+\sin\text{x}+3)+\mu(2\cos\text{x}+\sin\text{x}+3)+\text{v}$
$(5\cos\text{x}+6)=\lambda(-2\sin\text{x}+\cos\text{x})+\mu(2\cos\text{x}+\sin\text{x}+3)+\text{v}$
$(5\cos\text{x}+6)=(-2\lambda+\mu)\sin\text{x}(\lambda+2\mu)\cos\text{x}+(3\mu+\text{v})$
Comparing the coefficient of $\sin\text{x}\ \&\cos\text{x}$ on the both the sides,
$-2\lambda+\mu=0\dots\dots(1)$
$\lambda+2\mu=5\dots\dots(2)$
$3\mu+\text{v}=6\dots\dots(3)$
Solving equations (1), (2) and (3),
$\text{I}=\int\frac{(-2\sin\text{x}+\cos\text{x})}{(2\cos\text{x}+\sin\text{x}+3)}\text{dx}+2\int\text{dx}$
$\text{I}=\log|2\cos\text{x}+\sin\text{x}+3|+2\text{x}+\text{C}$

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 "5 Marks Questions" 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

Evaluate the definite integral in Exercise:
$\int^{\pi}_{0}\frac{\text{x}\tan\text{x}}{\sec\text{x}+\tan\text{x}}\text{dx}$

Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}=\frac{1+\text{y}^2}{\text{y}^3}$

Solve the following differential equations:
$(\text{x}^2-\text{y}^2)\text{dx}-2\text{xy dy}=0$

A factory has three machines A, B and C, which produce 100, 200 and 300 items of a particular type daily. The machines produce 2%, 3% and 5% defective items respectively. One day when the production was over, an item was picked up randomly and it was found to be defective. Find the probability that it was produced by machine A.

Evaluate the following integrals:
$\int\limits^2_{-1}\big(|\text{x}+1|+|\text{x}|+|\text{x}-1|\big)\text{dx}$

Find the intervals in which $\text{f(x)} = \sin 3x - \cos 3x, 0 < x < \pi.$ is strictly increasing or strictly decreasing.

Find the points of discontinuity, if any of the following function:
$\text{f(x)}=\begin{cases}\text{x}^3-\text{x}^2+2\text{x}-2,&\text{if }\text{ x}\neq1\\4,&\text{if }\text{ x}=1\end{cases}$

Find the distance of the point (-1, -5, -10) from the point of intersection of the line $\vec{\text{r}}=2\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}}+\lambda\Big(3\hat{\text{i}}+4\hat{\text{j}}+2\hat{\text{k}}\Big)$ and the plane $\vec{\text{r}}.\Big(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\Big)=5.$

Evaluate the following integrals:
$\int\limits^\pi_0\sin^{100}\text{x}\cos^{101}\text{x dx}$

Evaluate the following integrals:
$\int\text{cosec}^43\text{x}\text{ dx}$