1- 2x dx - Vidyadip

Question

$\int\sin\text{x}\sqrt{1-\cos2\text{x}}\text{ dx}$

✓

Answer

Let I $=\int\sin\text{x}\sqrt{1-\cos2\text{x}}\text{ dx}.$ Then,
$\text{I}=\int\sin\text{x}\times\sqrt{2\sin^2\text{x}}\times\text{dx}$
$=\int\sin\text{x}\times\sqrt{2}\times\sin\text{x dx}$
$=\sqrt{2}\int\sin^2\text{x}\times\text{dx}$
$=\frac{\sqrt{2}}{2}\int2\sin^2\text{x}\times\text{dx}$
$=\frac{\sqrt{2}}{2}\Big[\text{x}-\frac{\sin2\text{x}}{2}\Big]+\text{C}$
$=\frac{\sqrt{2}\text{x}}{2}-\frac{\sqrt{2}}{4}\times\sin2\text{x}+\text{C}$
$=\frac{1}{\sqrt{2}}\times\text{x}-\frac{\sin2\text{x}}{2\sqrt{2}}+\text{C}$
$\therefore\text{I}=\frac{1}{\sqrt{2}}\times\text{x}-\frac{\sin2\text{x}}{2\sqrt{2}}+\text{C}$

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 "5 Marks Questions" from Indefinite Integrals→Indefinite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Evalute the following integrals:
$\int\frac{\cos4\text{x}-\cos2\text{x}}{\sin4\text{x}-\sin2\text{x}}\text{dx}$

Find the equation of the plane through the points $(2, 2, -1)$ and $(3, 4, 2)$ and parallel to the line whose direction ratios are $7, 0, 6.$

Differentiate the function $x^{x \cos x}+\frac{x^{2}+1}{x^{2}-1} w.r.t. x.$

If $\text{y}=\text{a}\{\text{x}+\sqrt{\text{x}^2+1}\}^\text{n}+\text{b}\{\text{x}-\sqrt{\text{x}^2+1}\}^{-\text{n},}$ prove that $(\text{x}^2-1)\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{x}\frac{\text{dy}}{\text{dx}}-\text{n}^2\text{y}=0.$

A firm manufacturing two types of electric items, $A$ and $B,$ can make a profit of $Rs. 20$ per unit of $A$ and $Rs. 30$ per unit of $B.$ Each unit of $A$ requires $3$ motors and $4$ transformers and each unit of $B$ requires $2$ motors and $4$ transformers. The total supply of these per month is restricted to $210$ motors and $300$ transformers. Type $B$ is an export model requiring a voltage stabilizer which has a supply restricted to $65$ units per month. Formulate the linear programing problem for maximum profit and solve it graphically.

Find the distance of the point (2, 12, 5) from the point of intersection of the line $\vec{\text{r}}=2\hat{\text{i}}-4\hat{\text{j}}+2\hat{\text{k}}+\lambda(3\hat{\text{i}}+4\hat{\text{j}}+2\hat{\text{k}})$ and $\vec{\text{r}}\cdot(\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}})=0.$

Find the area of the ragion bounded by $x^2+ 16y = 0$ and its latusrectum.

If lines $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4} \text{and} \frac{x - 3}{1} = \frac{y - k}{2} = \frac{z}{1}$ intersect, then find the value of k and hence find the equation of the plane containing these lines.

Examine the consistency of the system of equation $3x - y - 2z = 2;\,\,2y - z = - 1;3x - 5y = 3$

Differentiate the following functions with respect to x:
$(1+\cos\text{x})^\text{x}$