Evaluate the following intregals: 1 5-4 dx - Vidyadip

Question

Evaluate the following intregals:
$\int\frac{1}{5-4\sin\text{x}}\ \text{dx}$

✓

Answer

Let $\text{I}=\int\frac{1}{5-4\sin\text{x}}\ \text{dx}$
Put $\sin\text{x}=\frac{2\tan^\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}$
$\text{I}=\int\frac{1}{5-4\Bigg(\frac{2\tan\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}\Bigg)}\ \text{dx}$
$=\int\frac{1+\tan^2\frac{\text{x}}{2}}{5\Big(1+\tan^2\frac{\text{x}}{2}\Big)-4\Big(2-\tan\frac{\text{x}}{2}\Big)}\ \text{dx}$
$=\int\frac{\sec^2\frac{\text{x}}{2}}{5+5\tan^2\frac{\text{x}}{2}-8\tan\frac{\text{x}}{2}}\ \text{dx}$
Let $\tan\frac{\text{x}}{2}=\text{t}$
$\frac{1}{2}\sec^2\frac{\text{x}}{2}\text{dx}=\text{dt}$
$=\int\frac{2\text{dt}}{5\text{t}^2-8\text{t}+5}$
$=\frac{2}{5}\int\frac{\text{dt}}{\text{t}^2-2\text{t}\Big(\frac{4}{5}\Big)+\Big(\frac{4}{5}\Big)^2-\Big(\frac{4}{5}\Big)^2+1}$
$\text{I}=\frac{2}{5}\int\frac{\text{dt}}{\Big(\text{t}-\frac{4}{5}\Big)^2+\Big(\frac{3}{5}\Big)^2}$
$=\frac{2}{5}\times\frac{1}{\frac{3}{5}}\Bigg(\frac{\text{t}-\frac{4}{5}}{\frac{3}{5}}\Bigg)+\text{C}$
$=\frac{2}{3}\tan^{-1}\Big(\frac{5\text{t}-4}{3}\Big)+\text{C}$
$\text{I}=\frac{2}{3}\tan^{-1}\Big(\frac{5\tan\frac{\text{x}}{2}-4}{3}\Big)+\text{C}$

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 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

Evaluate the following:
$\int\frac{\text{x}}{\sqrt{\text{x}}+1}\text{dx}$
Hint: Put $\sqrt{\text{x}}=\text{z}$

Evaluate the following integrals:
$\int\limits^{\text{a}}_{-\text{a}}\frac{1}{1+\text{a}^{\text{x}}}\text{ dx},\text{ a}>0$

A bag contains 3 red and 2 black balls. One ball is drawn from it at random. Its colour is noted and then it is put back in the bag. A second draw is made and the same procedure is repeated. Find the probability of drawing,

Two red balls,
Two black balls,
First red and second black ball.

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

If $\text{A} = \begin{bmatrix}2&-3&5\\3&2&-4\\1&1&-2\end{bmatrix},$ find A^-1. Using A^-1 solve the system of equations:

2x - 3y + 5z = 11

3x + 2y - 4z = -5

x + y - 2z = -3

Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}\sin^223^{\circ}&\sin^267^{\circ}&\cos180^{\circ}\\-\sin^267^{\circ}&-\sin^223^{\circ}&\cos^2180^{\circ}\\\cos180^{\circ}&\sin^223^{\circ}&\sin^267^{\circ}\end{vmatrix}$

Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat} defined as f(1) = a, f(2) = b, f(3) = c, g(a) = apple, g(b) = ball and g(c) = cat. Show that f, g and gof are invertible. Find f^-1, g^-1 and gof^-1 and show that (gof)^-1 = f^-1og^-1.

Solve the following differential equation:
$(1+\text{x}^2)\frac{\text{dy}}{\text{dx}}-2\text{xy}=(\text{x}^2+2)(\text{x}^2+1)$

Show that the differential equation $(x-y) \frac{d y}{d x}=x+2 y$ is homogeneous and solve it.

There are three categories of students in a class of 60 students:

A : Very hardworking

B : Regular but not so hardworking

C : Careless and irregular 10 students are in category A, 30 in category B and the rest in category C.

It is found that the probability of students of category A, unable to get good marks in the final year examination is 0.002, of category B it is 0.02 and of category C, this probability is 0.20. A student selected at random was found to be one who could not get good marks in the examination. Find the probability that this student is category C.