Integrate the function: x x x - Vidyadip

Question

Integrate the function: $\frac{\sqrt{\tan x}}{\sin x \cos x}$

✓

Answer

Let $I=\int \frac{\sqrt{\tan x}}{\sin x \cos x}$
$=\int \frac{\sqrt{\tan x} \cdot \cos x}{\sin x \cos x \cdot \cos x} d x$
$= \int \frac{\sqrt{\tan x}}{\tan x \cos ^{2} x} d x$
$= \int \frac{\sec ^{2} x d x}{\sqrt{\tan x}}$
Let tan x = t $\Rightarrow$ sec²x dx = dt
$\Rightarrow I=\int \frac{d t}{\sqrt{t}}$
$=2 \sqrt{t}+C$
$=2 \sqrt{\tan x}+c$

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

Find the area of the region bounded by the curve y = x - 1 and (y - 1)² = 4(x + 1).

Find the general solution of $(\text{x}+2\text{y}^3)\frac{\text{dy}}{\text{dx}}=\text{y}.$

Two sides of a triangle have lengths 'a' and 'b' and the angle between them is θ. What value of θ will maximize the area of the triangle? Find the maximum area of the triangle also.

If $\vec{\text{a}}$ and $\vec{\text{b}}$ are two non-collinear unit vectors such that $\big|\vec{\text{a}}+\vec{\text{b}}\big|=\sqrt{3},$ find $\big(2\vec{\text{a}}-5\vec{\text{b}}\big).\big(3\vec{\text{a}}+\vec{\text{b}}\big).$

Prove that the area common to tha two parabola y = 2x² and y = x² + 4 is 323 sq. units.

Find the matrix X for which:

$\begin{bmatrix}3 & 2 \\ 7 & 5 \end{bmatrix}\text{X}\begin{bmatrix} -1 & 1 \\ -2 & 1 \end{bmatrix}=\begin{bmatrix} 2 & -1 \\ 0 & 4 \end{bmatrix}$

Find the vector equation of the line passing through the points (-1, 0, 2) and (3, 4, 6).

If A + B + C = 0, then prove that $\begin{vmatrix}1&\cos\text{C}&\cos\text{B}\\\cos\text{C}&1&\cos\text{A}\\\cos\text{B}&\cos\text{A}&1\end{vmatrix}=0.$

There are two types of fertilisers 'A' and 'B'. 'A' consists of 12% nitrogen and 5% phosphoric acid whereas 'B' consists of 4% nitrogen and 5% phosphoric acid. After testing the soil conditions, farmer finds that he needs at least 12kg of nitrogen and 12kg of phosphoric acid for his crops. If 'A' costs Rs. 10 per kg and 'B' cost Rs. 8 per kg, then graphically determine how much of each type of fertiliser should be used so that nutrient requiremnets are met at a minimum cost.

Evaluate :
$
\int_0^{\frac{\pi}{2}} \frac{\cos ^2 x}{1+3 \sin ^2 x} d x
$