Evaluate the following integrals: 1 x^3 ( )dx - Vidyadip

Question

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

✓

Answer

Let $\text{I}=\int\frac{1}{\text{x}^3}\sin(\log\text{x})\text{dx}$
Putting log x = t
$\Rightarrow\text{x}=\text{e}^\text{t}$
$\Rightarrow\text{dx}=\text{e}^\text{t}\text{dt}$
$\therefore\ \text{I}=\int\frac{1}{\text{e}^{3\text{t}}}\sin\text{t e}^\text{t}\text{dt}$
$=\int\text{e}^{-2\text{t}}\sin\text{t dt}$
Considering sin t as first function and e^-2t as second function
$\text{I}=\sin\text{t}\Big[\frac{\text{e}^{-2\text{t}}}{-2}\Big]-\int\cos\text{t}\frac{\text{e}^{-2\text{t}}}{-2}\text{dt}$
$\Rightarrow\text{I}=\frac{\sin\text{t e}^{-2\text{t}}}{-2}+\frac{1}{2}\int\cos\text{t e}^{-2\text{t}}\text{dt}$
$\Rightarrow\text{I}=\frac{\sin\text{t e}^{-2\text{t}}}{-2}+\frac{1}{2}\Big[\cos\text{t}\frac{\text{e}^{-2\text{t}}}{-2}-\int(-\sin\text{t})\frac{\text{e}^{-2\text{t}}}{-2}\text{dt}\Big]$
$\Rightarrow\text{I}=\frac{\sin\text{t e}^{-2\text{t}}}{-2}-\frac{1}{4}\cos\text{t e}^{-2\text{t}}-\int\frac{\text{e}^{-2\text{t}}\sin\text{t dt}}{4}$
$\Rightarrow\text{I}=\text{e}^{-2\text{t}}\Big[\frac{-2\sin\text{t}-\cos\text{t}}{4}\Big]-\frac{\text{I}}{4}$
$\Rightarrow\frac{5\text{I}}{4}=\text{e}^{-2\text{t}}\Big[\frac{-2\sin\text{t}-\cos\text{t}}{4}\Big]$
$\Rightarrow\text{I}=\frac{\text{e}^{-2\text{t}}}{5}[-2\sin\text{t}-\cos\text{t}]+\text{C}$
$\Rightarrow\text{I}=\frac{-\text{x}^{-2}}{5}[2\sin(\log\text{x})+\cos(\log\text{x})]+\text{C}$
$\Rightarrow\text{I}=\frac{-1}{5\text{x}^2}[\cos(\log\text{x})+2\sin(\log\text{x})]+\text{C}$

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 "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

Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection,
f(x) = (x - 1)(x - 2)²

Find the equation of a plane which is at a distance of $3\sqrt{3}\text{ units}$ from the origin and the normal to which is equally inclined to the coordinate axes.

Find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x²+ y² = 32.

$\text{Evaluate:}\int\frac{\text{x + 2}}{\sqrt{\text{x}^{2}+\text{2x}+}\text{3}}\text{dx}$

Solve the following systems of linear equations by cramer's rule:

Find the vector and cartesian equations of the line passing through the point (2, 1, 3) and perpendicular to the lines$\frac{\text{x}-1}{1}=\frac{\text{y}-2}{2}=\frac{\text{z}-3}{3}$ and$\frac{\text{x}}{-3}=\frac{\text{y}}{2}=\frac{\text{z}}{5}.$

Evaluate the following integrals:
$\int\limits^{\frac{3}{2}}_0\big|\text{x}\sin\pi\text{x}\big|\text{dx}$

Let X denot the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in x number of colleges. It is given that
$\text{P}(\text{X = x})=\begin{cases}\text{kx},&\text{if}\text{ x}=0\text{ or }1\\2\text{kx},&\text{if x = 2}\\\text{k}(5-\text{x}),&\text{if x = 3 or 4}\\0,&\text{if x > 4}\end{cases}$
where k is a positive constant. Find the value of k. Also find the probability that you will get addmission in

Exactly one college.
At most two colleges.
At least two colleges.

Find the equation of the plane through the point $2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}$ and passing throught the line of intersection of the planes $\vec{\text{r}}\cdot(\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}})=0$ and $\vec{\text{r}}\cdot(\hat{\text{j}}+2\hat{\text{k}})=0.$

Differentiate the following with respect to x:
$\sin^{-1}\bigg(\frac{2^{\text{x} + 1}.3^{\text{x}}}{1 + (36)^{\text{x}}}\bigg)$