Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals4 Marks
Question
Evaluate the following integrals:
$\int\tan\text{x}\sec^2\text{x}\sqrt{1-\tan^2\text{x}}\text{ dx}$
✓

Answer

Let $\text{I}=\int\tan\text{x}\sec^2\text{x}\sqrt{1-\tan^2\text{x}}\text{ dx}\ ....(1)$

Let $1-\tan^2\text{x}=\text{t}$ then,

$\Rightarrow\text{d}\big(1-\tan^2\text{x}\big)\text{dt}$

$\Rightarrow-2\tan\text{x}\sec^2\text{x}\text{ dx}=\text{dt}$

$\Rightarrow\tan\text{x}\sec^2\text{x}\text{ dx}=\frac{-\text{dt}}{2}$

Putting $1-\tan^2\text{x}=\text{t}$ and $\tan\text{x}\sec^2\text{x}\text{ dx}=-\frac{\text{dt}}{2}$ in equation (1),

We get

$\text{I}=\int\sqrt{\text{t}}\times\frac{-\text{dt}}{2}$

$=\frac{-1}{2}\int\text{t}^{\frac{1}{2}}\text{dt}$

$=-\frac{1}{2}\times\frac{\text{t}^{\frac{3}{2}}}{\frac{3}{2}}+\text{C}$

$=-\frac{1}{3}\text{t}^\frac{3}{2}+\text{C}$

$=-\frac{1}{3}\big[1-\tan^2\text{x}\big]^{\frac{3}{2}}+\text{C}$

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 "4 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

Evaluate the following integrals:
$\int\limits_{0}^{\frac{\pi}{4}}\frac{\tan^{3}\text{x}}{1+\cos2\text{x}}\text{ dx}$
→
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 1kg of food X and 1kg 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.
→
If  xy = ex –y, show that  $\frac{\text{dy}}{\text{dx}}=\frac{\text{log x}}{\left\{\text{log(x e)}\right\}^{2}}.$
→
Differentiate the function $x^{x^{2}-3}+(x-3)^{x^{2}}, \text { for } x>3$ w.r.t to x.
→
Differentiate the following functions with respect to x:
$\text{x}\sin2\text{x}+5^{\text{x}}+\text{k}^\text{k}+(\tan^2\text{x})^3$
→
Find the differential equation of the family of curve $\text{y}=\text{Ae}^\text{2x}+\text{Be}^{-2\text{x}},$ where A and B are arbitrary constants.
→
Write the minors and cofactors of element of the first column of the following matrices and hence evaluate the determinant in case:
$\text{A}=\begin{vmatrix}1&\text{a}&\text{bc}\\1&\text{b}&\text{ca}\\1&\text{c}&\text{ab} \end{vmatrix}$
→
Find the vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x - y + z = 0.
→
Verify Rolle's theorem for the following function on the indicated intervals

f(x) = x(x - 2)2 on the interval [0, 2]

→
If $\text{A}=\begin{bmatrix}2&3\\1&2\end{bmatrix}$ and $\text{I}=\begin{bmatrix}1&0\\0&1\end{bmatrix},$ then find $\lambda,\mu$ so that $\text{A}^2=\lambda\text{A}+\mu\text{I}$
→