Evaluvate the following intregals: 2 +3 3 +4 dx - Vidyadip

Question

Evaluvate the following intregals:
$\int\frac{2\sin\text{x}+3\cos\text{x}}{3\sin\text{x}+4\cos\text{x}}\ \text{dx}$

✓

Answer

Let $\text{I}=\int\frac{2\sin\text{x}+3\cos\text{x}}{3\sin\text{x}+4\cos\text{x}}\ \text{dx}$
Let $(2\sin\text{x}+3\cos\text{x})=\lambda\frac{\text{d}}{\text{dx}}(3\sin\text{x}+4\cos\text{x})+\mu(3\sin\text{x}+4\cos\text{x})+\text{v}$
$(2\sin\text{x}+3\cos\text{x})=\lambda(3\cos\text{x}-4\sin\text{x})+\mu(3\sin\text{x}+4\cos\text{x})+\text{v}$
$(2\sin\text{x}+3\cos\text{x})=(3\lambda+4\mu)\cos\text{x}+(-4\lambda+3\mu)\sin\text{x}+\text{v}$
Compairing the coefficient of $\sin\text{x},\cos\text{x}$ on both the sides,
$3\lambda+4\mu=3\dots\dots(1)$
$-4\lambda+3\mu=2\dots\dots(2)$
$\text{v}=0\dots\dots(3)$
Solving the equation (1), (2) and (3)
$\lambda=\frac{1}{25}$
$\mu=\frac{18}{25}$
$\text{v}=0$
$\text{I}=\frac{1}{25}\int\frac{(3\cos\text{x}-4\sin\text{x})}{(3\sin\text{x}+4\cos\text{x})}\text{dx}+\frac{18}{25}\int\text{dx}$
$\text{I}=\frac{1}{25}\log|3\sin\text{x}+4\cos\text{x}|+\frac{18}{25}\text{x}+\text{C}$

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 following:
$\begin{pmatrix}\begin{bmatrix}1&3\\-1&-4 \end{bmatrix}+\begin{bmatrix}3&-2\\-1&1 \end{bmatrix}\end{pmatrix}\begin{bmatrix}1&3&5\\2&4&6 \end{bmatrix}$

Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}2&3&7\\13&17&5\\15&20&12 \end{vmatrix}$

Find the equation of the plane through the intersection of the planes 3x - 4y + 5z = 10 and 2x + 2y - 3z = 4 and parallel to the line x = 2y = 3z.

Two institutions decided to award their employees for the three values of resourcefulness, competence and determination in the form of prices at the rate of Rs. x, y and z respectively per person. The first institution decided to award respectively 4, 3 and 2 employees with a total price money of Rs. 37000 and the second institution decided to award respectively 5, 3 and 4 employees with a total price money of Rs. 47000. If all the three prices per person together amount to Rs. 12000 then using matrix method find the value of x, y and z. What values are described in this equations?

Find the equation of tangents to the curve $\text{y} = \cos (x + y), -2\pi \leq x \leq 2\pi$ that are parallel to the line x + 2y = 0.

Evaluate the following integrals:$\int\frac{\text{x}^3-3\text{x}}{\text{x}^4+2\text{x}^2-4}\text{ dx}$

Show that the following set of curves intersect orthogonally.
$x^3 - 3xy^2 = -2$ and $3x^2y - y^3 = 2$

Show that the following systems of linear equations has infinite number of solutions and solve:

$2x + y - 2z = 0,$

$x - 2y + z = -2,$

$5x - 5y + z = -2$

Find the equation of a plane passing through the points $(0, 0, 0)$ and $(3, -1,2)$ and parallel to the line $\frac{\text{x}-4}{1}=\frac{\text{y}+3}{-4}=\frac{\text{z}+1}{7}$

Show that $\text{f}(\text{x})=\frac{1}{1+\text{x}^2}$ is neither increasing nor decreasing on R.