Evaluate the following integrals: 1 (5-4 ) dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{1}{\cos\text{x}(5-4\sin\text{x})}\ \text{dx}$

✓

Answer

Let $\text{I}=\int\frac{1}{\sin\text{x}\sin2\text{x}}\ \text{dx}$
$\int\frac{\sin\text{x dx}}{\sin^2\text{x}+2\sin\text{x}\cos\text{x}}$
$=\int\frac{\sin\text{x dx}}{(1-\cos^2\text{x})+2(1-\cos^2\text{x})\cos\text{x}}$
Let $\cos\text{x}=\text{t}$
$\Rightarrow-\sin\text{x dx}=\text{dt}$
$\therefore\text{I}=\int\frac{\text{dt}}{(\text{t}^2-1)+2(\text{t}^2-1)\text{t}}$
Let $\frac{1}{(\text{t}^2-1)(3+2\text{t})}=\frac{\text{A}}{\text{t}-1}+\frac{\text{B}}{\text{t}+1}+\frac{\text{C}}{1+2\text{t}}$
⇒ 1 = A(t + 1)(1 + 2t) + B (t - 1)(1 + 2t) + C (t² - 1)
Put t = 1
⇒ 1 = 6A ⇒ $\text{A}=\frac{1}{6}$
Put t = -1
⇒ 1 = 2B ⇒ $\text{B}=-\frac{1}{2}$
Put $\text{t}=-\frac{1}{2}$
$\Rightarrow1=-\frac{3}{4}\text{C}\Rightarrow\text{C}=-\frac{4}{3}$
thus,
$\text{I}=\frac{1}{6}\int\frac{\text{dt}}{\text{t}-1}+\frac{1}{2}\int\frac{\text{dt}}{\text{t}+1}-\frac{4}{3}\int\frac{\text{dt}}{1+2\text{t}}$
$=\frac{1}{6}\log|\text{t}-1|+\frac{1}{2}\log|\text{t}+1|-\frac{2}{3}\log|1+2\text{t}|+\text{}C$
Hence,
$\text{I}=\frac{1}{6}\log|\cos\text{x}-1|+\frac{1}{2}\log|\cos\text{x}+1|+\frac{2}{3}\log|1+2\cos\text{x}|+\text{C}$

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 "4 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

A furniture trader deals in only two items – chairs and tables. He has ₹ 50,000 to invest and a space to store at most 35 items. A chair costs him ₹ 1000 and a table costs him ₹ 2000 The trader earns a profit of ₹ 150 and ₹ 250 on a chair and table, respectively. Formulate the above problem as an LPP to maximise the profit and solve it graphically.

Show that the sum of three vectors determined by the medians of a triangle directed from the vertices is zero.

A card is drawn from a pack of 52 cards so that each card is equally likely to be selected. In which of the following cases are the events A and B independent?
A = the card drawn is a spade,
B = the card drawn in an ace.

Find the direction cosines of the line $\frac{4-\text{x}}{2}=\frac{\text{y}}{6}=\frac{1-\text{z}}{3}.$ Also, reduce it to vector form

Differentiate $\sin^{-1}\Big(2\text{x}\sqrt{1-\text{x}^2}\Big)$ with respect to $\tan^{-1}\Big(\frac{\text{x}}{\sqrt{1-\text{x}^2}}\Big),$ if $-\frac{1}{\sqrt{2}}<\text{x}<\frac{1}{\sqrt{2}}$

Find the coordinates of the point where the line $\frac{\text{x}-2}{3}=\frac{\text{y}+1}{4}=\frac{\text{z}-2}{2}$ intersect the plane x - y + z - 5 = 0. Also, find the angle between the line and the plane.

Solve the following differential equation:
$(\text{x}+\text{y}+1)\frac{\text{dy}}{\text{dx}} = 1$

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.

Find the equation of a plane passing through the point $\text{P (6, 5, 9)}$ and parallel to the plane determined by the points $\text{A (3, –1, 2), B (5, 2, 4)}$ and $\text{ C (–1, –1, 6)}$ . Also find the distance of this plane from the point A.

Find the inverse of the following matrices:

$\begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\alpha & \sin\alpha \\ 0 & \sin\alpha & -\cos\alpha \end{bmatrix}$