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

In a class, $5 \%$ of the boys and $10 \%$ of the girls have an IQ of more than $150$. In this class, $60 \%$ of the students are boys. If a student is selected at random and found to have an IQ of more than $150$, find the probability that the student is a boy.

Evaluate the following integrals:$\int\limits^4_{0}\big(|\text{x}|+|\text{x}+2|+|\text{x}+4|\big)\text{dx}$

Show that lines $\bar{r}=(2 \hat{j}-3 \hat{k})+\lambda(\hat{i}+2 \hat{j}+3 \hat{k})$ and

$\bar{r}=(2 \hat{i}+6 \hat{j}+3 \hat{k})+\mu(2 \hat{i}+3 \hat{j}+4 \hat{k})$ are coplanar. Find the equation of the plane

determined by them.

A coin is tossed 5 times. What is the probability of getting at least 3 heads?

Evaluate the following intregals:
$\int\frac{\text{x}}{(\text{x}^2-\text{a}^2)(\text{x}^2-\text{b}^2)}\ \text{dx}$

Maximum Z = 2x + 3y
Subject to
$\text{x}+\text{y}\geq1$
$10\text{x}+\text{y}\geq5$
$\text{x}+10\text{y}\geq1$
$\text{x},\text{y}\geq0$

find the area of the region included between the parabola $y^2 = x$ and the line $x + y = 2$.

Evaluate the following integrals:$\int_{0}^\limits{1}\frac{\tan^{-1}\text{x}}{1+\text{x}^2}\text{ dx}$

Find the equation of the perpendicular drawn from the point P(-1, 3, 2) to the line $\vec{\text{r}}=\big(2\hat{\text{j}}+3\hat{\text{k}}\big)+\lambda\big(2\hat{\text{i}}+\hat{\text{j}}+3\hat{\text{k}}\big).$ Also, find the coordinates of the foot of the perpendicular from P.

Prove the following results:
$2\sin^{-1}\frac{3}{5}=\tan^{-1}\frac{24}{7}$