Find the integrals of the function 1 x ^ 3 x - Vidyadip

Question

Find the integrals of the function $\frac{1}{\sin x \cos ^{3} x}$

✓

Answer

Clearly, $\frac{1}{\sin x \cos ^{3} x}=\frac{\sin x}{\cos ^{3} x}+\frac{1}{\sin x \cos x}$
$= \tan x \sec ^{2} x+\frac{\frac{1}{\cos ^{2} x}}{\frac{\sin x \cos x}{\cos ^{2} x}}$
$= \tan x \sec ^{2} x+\frac{\sec ^{2} x}{\tan x}$
Now, $\int \frac{1}{\sin x \cos ^{3} x} d x=\int \tan x \sec ^{2} x d x+\int \frac{\sec ^{2} x}{\tan x} d x$
Let tan x = t
$\Rightarrow \sec^2 x dx = dt$
$\Rightarrow \int \frac{1}{\sin x \cos ^{3} x} d x=\int \operatorname{td} t+\int \frac{1}{t} d t$
$= \frac{t^{2}}{2}+\log |t|+C$
$= \frac{1}{2} \tan ^{2} x+\log |\tan x|+C$

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 "3 Marks Question" 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

Evaluate the following:
$\tan^{-1}\Big(\tan\frac{5\pi}{6}\Big)+\cos^{-1}\Big\{\cos\Big(\frac{13\pi}{6}\Big)\Big\}$

Using properties of determinants, prove that:
$\begin{vmatrix}1&1+\text{p}&1+\text{p}+\text{q}\\2&3+2\text{p}&4+3\text{p}+\text{2q}\\3&6+3\text{p}&10+6\text{p}+3\text{q}\end{vmatrix}=1$

Dot product of a vector with $\hat{\text{i}}+\hat{\text{j}}-3\hat{\text{k}},\hat{\text{i}}+3\hat{\text{j}}-2\hat{\text{k}}$ and $2\hat{\text{i}}+\hat{\text{j}}+4\hat{\text{k}}$ are 0, 5 and 8 respectively. Find the vector.

Evalute the following integrals:
$\int\frac{\text{e}^{2\text{x}}}{\text{e}^{2\text{x}}-2}\text{dx}$

Three relation $R_3$ is defined in set $A=\{a, b, c\}$ as follows:
$R_3=\{(b, c)\}$
Find whether or not the relation $R_3$ on $A$ is:

Reflexive.
Symmetric.
Transitive.

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

Find fog and gof if:$f(x) = e^x, g(x) = log_ex$

If $f(x)$ is defined by $f(x) x^2$. find $f(2)$.

Find the equation of rthe planes parallel to the plane x + 2y - 2z + 8 = 0 that are at a distance of 2 units from the point (2, 1, 1).

Evaluate the following determinant:
$\begin{vmatrix}\cos15^\circ&\sin15^\circ\\\sin75^\circ&\cos75^\circ \end{vmatrix}$