Evaluate the following integrals: dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{\tan\text{x}}{\sqrt{\cos\text{x}}}\text{dx}$

✓

Answer

$\int\frac{\tan\text{x}}{\sqrt{\cos\text{x}}}\text{dx}$
$\Rightarrow\int\frac{\sin\text{x}}{\cos\text{x}\sqrt{\cos\text{x}}}\text{dx}$
$\Rightarrow\int\frac{\sin\text{x}}{\cos^\frac{3}{2}\text{x}}\text{dx}$
$\text{Let }\cos\text{x}=\text{t}$
$\Rightarrow-\sin\text{x}\text{ dx}=\text{dt}$
$\Rightarrow\sin\text{x}=-\frac{\text{dt}}{\text{dx}}$
$\text{Now,}\int\frac{\sin\text{x}}{\cos^\frac{3}{2}\text{x}}\text{dx}$
$=\int-\frac{1}{\text{t}^\frac{3}{2}}\text{dt}$
$=-\int\text{t}^{-\frac{3}{2}}\text{dt}$
$=-\bigg[\frac{\text{t}^{-\frac{3}{2}+1}}{\frac{-3}{2}+1}\bigg]+\text{C}$
$=\frac{2}{\sqrt{\text{t}}}+\text{C}$
$=\frac{2}{\sqrt{\cos\text{x}}}+\text{C}$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

If $\text{f}\text{(x)}=\begin{cases}\frac{\text{x}^2-1}{\text{x}-1},& \text{for }\text{ x}\neq1 \\2,&\text{for }\text{ x}=1\end{cases}$ Find whether f (x) is continuous at x = 1.

Given that the two numbers appearing on throwing two dice are different. Find the probability of the event 'the sum of numbers on the dice is 4'.

Evalute the following integrals:
$\int\frac{1}{\text{e}^\text{x}+1}\text{dx}$

$\begin{bmatrix}2&3\\5&7\end{bmatrix}\begin{bmatrix}1&-3\\-2&4\end{bmatrix}=\begin{bmatrix}-4&6\\-9&\text{x}\end{bmatrix}$ find x.

Evaluate the following integrals:
$\int\frac{1}{\text{x}^3}\sin(\log\text{x})\text{dx}$

Evaluate the following integrals:$\int\limits^{\pi}_0\frac{\text{x}}{1+\cos\alpha\sin\text{x}}\text{ dx},0<\alpha<\pi$

If $\text{y}=(\tan\text{x})^{(\tan\text{x})^{(\tan\text{x})^{....\infty}}},$ prove that $\frac{\text{dy}}{\text{dx}}=2\text{ at x}=\frac{\pi}{4}$

Show that the points $\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}$ and $3\hat{\text{i}}+3\hat{\text{j}}+3\hat{\text{k}}$ are equidistant from the plane $\vec{\text{r}}\cdot(5\hat{\text{i}}+2\hat{\text{j}}-7\hat{\text{k}})+9=0$

In a hospital, there are 20 kidney dialysis machines and that the chance of any one of them to be out of service during a day is 0.02. Determine the probability that exactly 3 machines will be out of service on the same day.

If $\vec{\text{p}}=5\hat{\text{i}}+\lambda\hat{\text{j}}-3\hat{\text{k}}$ and $\vec{\text{q}}=\hat{\text{i}}+3\hat{\text{j}}-5\hat{\text{k}},$ then find the value of $\lambda,$ so that $\vec{\text{p}}+\vec{\text{q}}$ and $\vec{\text{p}}-\vec{\text{q}}$ are perpendicular vectora.