Evaluate the following integrals: ^4xdx - Vidyadip

Question

Evaluate the following integrals:
$\int\tan\text{x }\sec^4\text{x}\text{dx}$

✓

Answer

Let $\text{I}=\int\tan\text{x }\sec^4\text{x}\text{dx}$ Then
$\text{I}=\int\tan\text{x }\sec^2\text{x}\sec^2\text{x}\text{dx}$
$=\int\tan\text{x}(1+\tan^2\text{x})\sec^2\text{x}\text{dx}$
$\text{I}=\int\big(\tan\text{x}+\tan^3\text{x}\big)\sec^2\text{x}\text{dx}$
Substituting $\tan\text{x}=\text{t}$ and $\sec^2\text{xdx}=\text{dt},$ we get
$\text{I}=\int(\text{t}+\text{t}^3)\text{dt}$
$=\frac{\text{t}^2}{2}+\frac{\text{t}^4}{4}+\text{C}$
$=\frac{\tan^2\text{x}}{2}+\frac{\tan^4}{4}+\text{C}$
$\therefore\ \text{I}=\frac{1}{2}\times\tan^2\text{x}+\frac{1}{4}\times\tan^4\text{x}+\text{C}$

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 "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

Express $\overrightarrow{\text{AB}}$ in terms of unit vectors $\hat{\text{i}}\text{ and }\hat{\text{j}}$, when the point is:

A(4, -1), B(1, 3)

Find $\Big|\overrightarrow{\text{AB}}\Big|$

Find the mean and standard deviation of the following probability distributions:

x_i	2	3	4
p_i	0.2	0.5	0.3

Integrate the function: $\left(x^{3}-1\right)^{\frac{1}{3}} x^{5}$

$\tan^{-1}\Bigg(\frac{\text{x - 1}}{\text{x - 2}}\Bigg)+\tan^{-1}\Bigg(\frac{\text{x + 1}}{\text{x + 2}}\Bigg)=\frac{\pi}{4}.$

Evaluate the following integrals:
$\int\limits^8_2|\text{x}-5|\text{dx}$

Find the slope of the tangent to the curve x = t² + 3t - 8, y = 2t² - 2t - 5 at t = 2.

Check whether the relation R on R defined by R = {(a, b): a ≤ b³} is reflexive, symmetric or transitive.

Find the mean μ variance σ² for the following probability distribution:

X	0	1	2	3
P(X)	$\frac{1}{6}$	$\frac{1}{2}$	$\frac{3}{10}$	$\frac{1}{30}$

By using the properties of definite integrals, evaluate the integral $\int\limits_{ - 5}^5 {\left| {x + 2} \right|dx} $

Find the unit vector in the direction of sum of vectors $\vec{\text{a}}=2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$ and $\vec{\text{b}}=2\hat{\text{j}}+\hat{\text{k}}.$