Evaluate the following: ^2x ^4x dx - Vidyadip

Question

Evaluate the following:
$\int\tan^2\text{x}\sec^4\text{x dx}$

✓

Answer

Let $\text{I}=\int\tan^2\text{x}\sec^4\text{xdx}$
$=\int\tan^2\text{x}\sec^2\text{x}\sec^2\text{xdx}$
$=\int\tan^2\text{x}(1+\tan^2\text{x})\sec^2\text{xdx}$
Put $\tan\text{x}=\text{t}\Rightarrow\sec^2\text{x}\text{dx}=\text{dt}$
$\therefore\ \text{I}=\int\text{t}^2(1+\text{t}^2)\text{dt}$
$=\int(\text{t}^2+\text{t}^4)\text{dt}=\frac{\text{t}^3}{3}+\frac{\text{t}^5}{5}+\text{C}$ $=\frac{\tan^5\text{x}}{5}+\frac{\tan^3\text{x}}{3}+\text{C}$

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 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

Of the students in a college, it is known that 60% reside in a hostel and 40% do not reside in hostel. Previous year results report that 30% of students residing in hostel attain A grade and 20% of ones not residing in hostel attain A grade in their annual examination. At the end of the year, one students is chosen at random from the college and he has an A grade. What is the probability that the selected student is a hosteler?

If f : A → B and g : B → C are onto functions, show that gof is a onto function.

Find the position vector of the mid-point of the vector joining the points $\text{P}\big(2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}}\big)$ and $\text{Q}\big(4\hat{\text{i}}+\hat{\text{j}}-2\hat{\text{k}}\big)$.

If $\begin{bmatrix}2&1&3\end{bmatrix}\begin{bmatrix}-1&0&-1\\-1&1&0\\0&1&1\end{bmatrix}\begin{bmatrix}1\\0\\-1\end{bmatrix}=\text{A},$ then find the value of A.

Find the equation of the tangent to the curve $\sqrt{\text{x}}+\sqrt{\text{y}}=\text{a},$ at the point $\Big(\frac{\text{a}^2}{4},\frac{\text{a}^2}{4}\Big).$

Find the vector equation of the plane with intercepts 3, -4 and 2 on x, y and z-axis respectively.

Sketch the graph of y = |x + 3| and evaluate $\int_{ - 6}^0 {\left| {x + 3} \right|dx} $

Find the angle between the line $\frac{\text{x}+1}{2}=\frac{\text{y}}{3}=\frac{\text{z}-3}{6}$ and the plane 10x + 2y - 11z = 3.

Evaluate the following integrals:
$\int\frac{1}{\sqrt{1-\text{x}^2}(\sin^{-1}\text{x})^2}\text{dx}$

Prove the following:

$\tan^{-1}\Bigg(\frac{1}{3}\Bigg)+\tan^{-1}\Bigg(\frac{1}{5}\Bigg)+\tan^{-1}\Bigg(\frac{1}{7}\Bigg)+\tan^{-1}\Bigg(\frac{1}{8}\Bigg)=\frac{\pi}{4}.$