Evaluate the following integrals: ^42x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\sec^42\text{x}\text{ dx}$

✓

Answer

$\int\sec^42\text{x}\text{ dx}$
$=\int\sec^22\text{x}.\sec^22\text{x}\text{ dx}$
$=\int(1+\tan^22\text{x}).\sec^22\text{x}\text{ dx}$
Let $\tan2\text{x}=\text{t}$
$\sec^22\text{x}.2\text{dx}=\text{dt}$
$\sec^22\text{x}.\text{dx}=\frac{\text{dt}}{2}$
Now, $\int(1+\tan^22\text{x}).\sec^22\text{x}\text{ dx}$
$=\frac{1}{2}\int(1+\text{t}^2)\text{dt}$
$=\frac{1}{2}\Big[\text{t}+\frac{\text{t}^3}{3}\Big]+\text{C}$
$=\frac{\text{t}}{2}+\frac{\text{t}^3}{6}+\text{C}$
$=\frac{\tan(2\text{x})}{2}+\frac{\tan^3(2\text{x})}{6}+\text{C}$

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 "3 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

Find the second order derivatives of the following functions:
$\text{y}=\text{x}^3\log\text{x}$

Write a value of $\int(\text{e}^{\text{x}\log_\text{e}\text{a}}+\text{e}^{\text{a}\log_\text{e}\text{x}})\text{dx}$

Find the coordinates of the foot of the perpendicular and its length drawn from the point $P(5,4,2)$ to the line
$
\vec{r}=-\hat{i}+3 \hat{j}+\hat{k}+\lambda \quad(2 \hat{i}+3 \hat{j}-\hat{k})
$

If f : R → R be defined by f(x) = x³ - 3, then prove that f^-1 exists and find a formula for f^-1. Hence, find f^-1(24) and f^-1(5).

For each of the differential equations given in find the general solution:
$(\text{x}+3\text{y}^2)\frac{\text{dy}}{\text{dx}}=\text{y}(\text{y}>0)$

Evaluate the following integrals:
$\int\frac{\text{dx}}{\text{e}^{\text{x}}+\text{e}^{-\text{x}}}$

Two dice are thrown. Find the probability that the numbers appeared has the sum 8, if it is known that the second die always exhibits 4.

Evaluate the following integrals:

$\int\frac{\text{x}}{\sqrt{\text{x}^4+\text{a}^4}}\text{ dx}$

For each of the differential equations in find a particular solution satisfying the given condition:
$\frac{\text{dy}}{\text{dx}}=\text{y}$ tan x; y = 1 when x = 0

Solve the following differential equations:

$2\text{x}\frac{\text{dy}}{\text{dx}}=3\text{y},\text{y}(1)=2$