Evalute the following integrals: 2+3 dx - Vidyadip

Question

Evalute the following integrals:
$\int\frac{\cos\text{x}}{2+3\sin\text{x}}\text{dx}$

✓

Answer

Let $\text{I}=\int\frac{\cos\text{x}}{2+3\sin\text{x}}\text{dx}\ .....\text{(i)}$
Let $2+3\sin\text{x}=\text{t}$ then,
$\text{d}(2+3\sin\text{x})=\text{dt}$
$\Rightarrow3\cos\text{x dx}=\text{dt}$
$\Rightarrow\text{dx}=\frac{\text{dt}}{3\cos\text{x}}$
Putting $2+3\sin\text{x}=\text{t and dx}=\frac{\text{dt}}{3\cos\text{x}}$ in equation (i), we get,
$\text{I}=\int\frac{\cos\text{x}}{\text{t}}\times\frac{\text{dt}}{3\cos\text{x}}$
$=\frac{1}{3}\int\frac{\text{dt}}{\text{t}}$
$=\frac{1}{3}\log|\text{t}|+\text{C}$
$=\frac{1}{3}\log|2+3\sin\text{x}|+\text{C}$

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 Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Show that the line $\frac{\text{x}}{\text{a}}+\frac{\text{y}}{\text{b}}=1,$ touches the curve $\text{y}=\text{b}\cdot\text{e}^{\frac{-\text{x}}{\text{a}}}$ a e at the point where the curve intersects the axis of y.

show that the differential equation of which $\text{y}=2(\text{x}^2-1)+\text{ce}^{-\text{x}^{2}}$ is a solution, is $\frac{\text{dy}}{\text{dx}}+2\text{xy}=4\text{x}^3$

Show that $\text{A}=\begin{bmatrix} 5 & 3 \\-1 & -2 \end{bmatrix}$ satisfies the equation $x^2 - 3x - 7 = 0$. Thus, find $A^{-1}$.

Using integration, find the area of the following region:
$\Big\{(\text{x},\text{y}):\frac{\text{x}^2}{9}+\frac{\text{y}^2}{4}\leq1\leq\frac{\text{x}}{3}+\frac{\text{y}}{2}\Big\}$

Find all point of discontinuity of the function $\text{f(t)}=\frac{1}{\text{t}^2+\text{t}-2},$ where $\text{t}=\frac{1}{\text{x}-1}$

If the area bounded by the parabola $y^2 = 4ax$ and the line y = mx is $\frac{\text{a}^2}{12}\text{ sq.}$ units, then using integration, find the value of m.

Obtain the inverse of the following matrix using elementary operations: $\text{A}=\begin{bmatrix}-1 & 1 & 2\\1 & 2 & 3\\3 & 1 & 1 \end{bmatrix}$

Show that the four points (0, -1, -1), (4, 5, 1), (3, 9, 4) and (-4, 4, 4) are coplanar and find the equation of the common plane.

Find the angle between the lines whose direction ratios are proportional to a, b, c and b - c, c - a, a - b.

Find the slopes of the tangent and the normal to the following curves at the indicated points:
$\text{y}=(\sin2\text{x}+\cot\text{x}+2)^2\text{at}\text{ x}=\frac{\pi}{2}$