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 "5 Marks Questions" 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

At what points on the following curves, is the tangent parallel to x-axis?
$y = 12(x + 1)(x - 2)$ on $[-1, 2]$

Find the intervals in which the function $\text{f(x)}=\frac{\text{x}^4}{4}-\text{x}^3-5\text{x}^2+24\text{x}+12$ is (a) strictly increasing, (b) strictly decreasing.

Evaluate the following definite integrals:
$\int\limits_{0}^{1}\Big(\text{xe}^{\text{x}}+\cos\frac{\pi\text{x}}{4}\Big)\text{dx}$

Find the mean variance and standard deviation of the following probability distribution

$x_i$	a	b
$p_i$	p	q

Where $p + q = 1$

Compute the adjoint of the following matrices:$\begin{bmatrix}2 & -1 & 3 \\4 & 2 & 5 \\ 0 & 4 & -1 \end{bmatrix}$
Verify that (adj A)A = |A|I = A (adj A) for the above matrices.

Evaluate the following integrals:$\int\frac{\sqrt{1-\sin\text{x}}}{1+\cos\text{x}}\text{e}^{\frac{-\text{x}}{2}}\text{dx}$

To maintain one's health, a person must fulfil certain minimum daily requirements for the following three nutrients: calcium, protein and calories. The diet consists of only items I and II whose prices and nutrient contents are shown below:

	Food I	Food II	Minimum daily requirement
Calcium	10	4	20
Protein	5	6	20
Calories	2	6	12
Price	Rs. 0.60 per unit	Rs. 1.00 per unit

Find the combination of food items so that the cost may be minimum.

Evaluate the following integrals:
$\int\tan\text{x}\sec^2\text{x}\sqrt{1-\tan^2\text{x}}\text{ dx}$

Evaluate the following determinant:
$\begin{vmatrix}6&-3&2\\2&-1&2\\-10&5&2 \end{vmatrix}$

A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.