Evaluate the following integrals: ( 2x-1 2x+1 )dx - Vidyadip

Question

Evaluate the following integrals:
$\int\text{x}\Big(\frac{\sec2\text{x}-1}{\sec2\text{x}+1}\Big)\text{dx}$

✓

Answer

Let $\text{I}=\int\text{x}\Big(\frac{\sec2\text{x}-1}{\sec2\text{x}+1}\Big)\text{dx}$
$=\int\text{x}\Big(\frac{1-\cos2\text{x}}{1+\cos2\text{x}}\Big)\text{dx}$
$=\int\text{x}\Big(\frac{\sec^2\text{x}}{\cos^2\text{x}}\Big)\text{dx}$
$=\int\text{x}\tan^2\text{x dx}$
$=\int\text{x}(\sec^2\text{x}-1)\text{dx}$
$=\int\text{x}\sec^2\text{x dx}-\int\text{dx}$
$=\big[\text{x}\int\sec^2\text{x dx}-\int(1\int\sec^2\text{x dx})\text{dx}\big]-\frac{\text{x}^2}{2}$
$=\text{x}\tan\text{x}-\int\tan\text{x dx}-\frac{\text{x}^2}{2}$
$\text{I}=\text{x}\tan\text{x}-\log\sec\text{x}-\frac{\text{x}^2}{2}+\text{C}$

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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

An automobile manufacturer makes automobiles and trucks in a factory that is divided into two shops. Shop A, which performs the basic assembly operation, must work 5 man-days on each truck but only 2 man-days on each automobile. Shop B, which performs finishing operations, must work 3 man-days for each automobile or truck that it produces. Because of men and machine limitations, shop A has 180 man-days per week available while shop B has 135 man-days per week. If the manufacturer makes a profit of Rs 30000 on each truck and Rs 2000 on each automobile, how many of each should he produce to maximize his profit? Formulate this as a LPP.

Find the equation of the plane passing through the intersection of the planes $2x + 3y - z + 1 = 0$ and $x + y - 2z + 3 = 0$ and perpendicular to the plane $3x - y - 2z - 4 = 0.$

Form the differential equation of the family of curves represented by the equation (a being the perimeter):$(2\text{x}+\text{a})^2+\text{y}^2=\text{a}^2$

Let S be the set of all real numbers except -1 and let '*' be an operation defined by a * b = a + b + ab for all a, b ∈ S. Determine whether '*' is a binary operation on S. If yes, check its commutativity and associativity. Also, solve the equation (2 * x) * 3 = 7.

Differentiate the following functions with respect to x:
$\log(\text{x}+\sqrt{\text{x}^2+1})$

If the area bounded by the parabola $\text{y}^{2} = 16\text{ax}$ and the line $\text{y = 4 mx}$ is $\frac{\text{a}^{2}}{12}$ sq. units, then using integration, find the value of m.

$A$ discrete random variable $X$ has the probability distribution given as below:

$X$	$0.5$	$1$	$1.5$	$2$
$P(X)$	$k$	$k^2$	$2k^2$	$k$

Find the value of $k$.
Determine the mean of the distribution.

Solve the following differential equation
$\text{x}\cos\text{y dy}=(\text{xe}^\text{x}\log\text{x}+\text{e}^\text{x})\text{dx}$

A dietician mixes together two kinds of food in such a way that the mixture contains at least $6$ units of vitamin $A, 7$ units of vitamin $B, 11$ units of vitamin $C$ and $9$ units of vitamin $D.$ The vitamin contents of $1\ kg$ of food $X$ and $1\ kg$ of food $Y$ are given below:

	Vitamin $A$	Vitamin $B$	Vitamin $C$	Vitamin $D$
Food $X$	$1$	$1$	$1$	$2$
Food $Y$	$2$	$1$	$3$	$1$

One $kg$ food $X$ costs $Rs. 5,$ whereas one $kg$ of food $Y$ costs $Rs. 8.$
Find the least cost of the mixture which will produce the desired diet.

Find the integral of the function $\sin^3 (2x + 1)$