Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS4 Marks
Question
Evaluate the following integrals:
$\int^\limits{\frac{\pi}{6}}_{0}\cos^{-3}2\theta\sin2\theta\text{ d}\theta$
✓

Answer

We have,
$\int^\limits{\frac{\pi}{6}}_{0}\cos^{-3}2\theta\sin2\theta\text{ d}\theta$
$=\int^\limits{\frac{\pi}{6}}_{0}\frac{\sin2\theta}{\cos^32\theta}\text{ d}\theta$
$=\int^\limits{\frac{\pi}{6}}_{0}\tan2\theta\cdot\sec^22\theta\text{ d}\theta$
Let $\tan2\theta=\text{t}$
Differentiating w.r.t. x, we get
$2\sec^22\theta\text{d}\theta=\text{dt}$
Now, $\theta=0\Rightarrow\text{t}=0$
$\theta=\frac{\pi}{6}\Rightarrow\text{t}=\sqrt{3}$
$\therefore\ \int^\limits{\frac{\pi}{6}}_{0}\tan2\theta\cdot\sec^22\theta\text{ d}\theta=\frac{1}{2}\int^\limits{\sqrt{3}}_0\text{t dt}=\frac{1}{2}\Big[\frac{\text{t}^2}{2}\Big]^{\sqrt{\text{3}}}_0$
$=\frac{3}{4}$

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 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

Using properties of determinants, show that

$\Delta=\begin{vmatrix} \text{b+c} & \text{a} & \text{a} \\ \text{b} & \text{c+a} & \text{b} \\ \text{c} & \text{c} & \text{a+b} \end{vmatrix}=\text{4 abc}.$

→
A small manufacturer has employed 5 skilled men and 10 semi-skilled men and makes an article in two qualities deluxe model and an ordinary model. The making of a deluxe model requires 2 hrs. work by a skilled man and 2 hrs. work by a semi-skilled man. The ordinary model requires 1 hr by a skilled man and 3 hrs. by a semi-skilled man. By union rules no man may work more than 8 hrs per day. The manufacturers clear profit on deluxe model is Rs. 15 and on an ordinary model is Rs. 10. How many of each type should be made in order to maximize his total daily profit.
→
Find the Cartesian equation of the plane passing through the points A(0, 0, 0) and B(3, –1, 2) and parallel to the line $\frac{\text{x-4}}{1}=\frac{\text{y+3}}{-4}=\frac{\text{z+1}}{7}$ .
→
A diet of two foods F1 and F2 contains nutrients thiamine, phosphorous and iron.
The amount of each nutrient in each of the food (in milligrams per 25gms) is given in the following table:
Nutrients Food F1 F2
Thiamine 0.25 0.10
Phosphorous 0.75 1.50
Iron 1.60 0.80
The minimum requirement of the nutrients in the diet are 1.00mg of thiamine, 7.50mg of phosphorous and 10.00mg of iron.
The cost of F1 is 20 paise per 25gms while the cost of F2 is 15 paise per 25gms.
Find the minimum cost of diet.
→
Evaluate the following integrals:
$\int\limits_{0}^{1}\frac{1}{1+2\text{x}+2\text{x}^2+2\text{x}^3+\text{x}^4}\text{ dx}$
→
Differentiate the following functions with respect to x:
$\cos^{-1}\Big(\frac{\text{x}+\sqrt{1-\text{x}^2}}{\sqrt{2}}\Big),-1<\text{x}<1$
→
Find the absolute maximum and minimum values of a function f given by
$\text{f}(\text{x})=12\text{x}^\frac{4}{3}-6\text{x}^\frac{1}{3},\text{x}\in[-1,1]$
→
Let $\text{A}=\begin{bmatrix}3 & 2 \\7 & 5 \end{bmatrix}\text{and B}=\begin{bmatrix}6 & 7 \\8 & 9 \end{bmatrix}$. Find (AB)-1.
→

Maximize Z = 3x1 + 4x2, if possible,

Subject to the constraints

$\text{x}_1-\text{x}_2\leq-1$

$-\text{x}_1+\text{x}_2\leq0$

$\text{x}_1,\text{x}_2\geq0$

→
Prove that the line of section of the planes 5x + 2y - 4z + 2 = 0 and 2x + 8y + 2z - 1 = 0 is parallel to the plane 4x - 2y - 5z - 2 = 0.
→