Question
Evaluate the following integrals:$\int\text{e}^{\text{x}}\Big(\frac{\sin4\text{x}-4}{1-\cos4\text{x}}\Big)\text{dx}$
✓
Answer
Let $\text{I}=\int\text{e}^{\text{x}}\Big(\frac{\sin4\text{x}-4}{2\sin^22\text{x}}\Big)\text{dx}$
$=\int\text{e}^{\text{x}}\Big\{\frac{2\sin2\text{x}\cos2\text{x}}{2\sin^22\text{x}}-\frac{4}{2\sin^{2}2\text{x}}\Big\}\text{dx}$
$=\int\text{e}^{\text{x}}\big(\cot2\text{x}-2\text{cosec}^22\text{x}\big)\text{dx}$
$=\int\text{e}^{\text{x}}\cot2\text{x dx}-2\int\text{e}^{\text{x}}\text{cosec}^22\text{x dx}$
integrating by parts
$=\text{e}^{\text{x}}\cot2\text{x}-\int\text{e}^{\text{x}}\frac{\text{d}}{\text{dx}}(\cot2\text{x})\text{dx}-2\int\text{e}^{\text{x}}\text{cosec}^22\text{x dx}$
$=\text{e}^{\text{x}}\cot2\text{x}+2\int\text{e}^{\text{x}}\text{cosec}^22\text{x}-2\int\text{e}^{\text{x}}\text{cosec}^22\text{x dx}$
$=\text{e}^{\text{x}}\cot2\text{x}+\text{C}$
$=\int\text{e}^{\text{x}}\Big\{\frac{2\sin2\text{x}\cos2\text{x}}{2\sin^22\text{x}}-\frac{4}{2\sin^{2}2\text{x}}\Big\}\text{dx}$
$=\int\text{e}^{\text{x}}\big(\cot2\text{x}-2\text{cosec}^22\text{x}\big)\text{dx}$
$=\int\text{e}^{\text{x}}\cot2\text{x dx}-2\int\text{e}^{\text{x}}\text{cosec}^22\text{x dx}$
integrating by parts
$=\text{e}^{\text{x}}\cot2\text{x}-\int\text{e}^{\text{x}}\frac{\text{d}}{\text{dx}}(\cot2\text{x})\text{dx}-2\int\text{e}^{\text{x}}\text{cosec}^22\text{x dx}$
$=\text{e}^{\text{x}}\cot2\text{x}+2\int\text{e}^{\text{x}}\text{cosec}^22\text{x}-2\int\text{e}^{\text{x}}\text{cosec}^22\text{x dx}$
$=\text{e}^{\text{x}}\cot2\text{x}+\text{C}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
A manufacturer makes two types $A$ and $B$ of tea-cups. Three machines are needed for the manufacture and the time in minutes required for each cup on the machines is given below:
Each machine is available for a maximum of $6$ hours per day. If the profit on each cup $A$ is $75$ paise and that on each cup $B$ is $50$ paise, show that $15$ tea-cups of type $A$ and $30$ of type $B$ should be manufactured in a day to get the maximum profit.
→| Machines | |||
| $I$ | $II$ | $III$ | |
| $A$ | $12$ | $18$ | $6$ |
| $B$ | $6$ | $0$ | $9$ |
$\text{Let f : N}\rightarrow\text{N}$ be a function defined as $\text{f(x)} = 9x^{2} + 6x -5.$ Show that $\text{f : N}\rightarrow\text{S},$ where S is the range of f, is invertible. Find the inverse of f and hence find $\text{f}^{-1}(43) \text{and f}^{-1}(163). $
→Compute the adjoint of the following matrices:
$\begin{bmatrix} 1 & 2 & 5 \\ 2 & 3 & 1 \\ -1 & 1 & 1 \end{bmatrix}$
Verify that (adjoint A)A = |A|I = A (adjoint A) for the above matrices.
→$\begin{bmatrix} 1 & 2 & 5 \\ 2 & 3 & 1 \\ -1 & 1 & 1 \end{bmatrix}$
Verify that (adjoint A)A = |A|I = A (adjoint A) for the above matrices.
Show that the vector $\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ is equally inclined to the coordinate axes.
→Solve the following systems of linear equations by cramer's rule:
2x - 3y - 4z = 29,
-2x + 5y - z = -15,
3x - y + 5z = -11
Evaluate the following integrals:
$\int_{0}^\limits{\frac{\pi}{4}}\big(\text{a}^2\cos^2\text{x}+\text{b}^2\sin^2\text{x}\big)\text{dx}$
→$\int_{0}^\limits{\frac{\pi}{4}}\big(\text{a}^2\cos^2\text{x}+\text{b}^2\sin^2\text{x}\big)\text{dx}$
Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix}5 & 2 \\ 2 & 1 \end{bmatrix}$
→Find one-parameter families of solution curves of the following differential equation: (or solve the following differential equation)$\frac{\text{dy}}{\text{dx}}+3\text{y}=\text{e}^{\text{mx}},$ m is given real number.
→Let A be the set of all human beings in a town at a particular time. Determine whether the following relations are reflexive, symmetric and transitive:
R = {(x, y): x and y work at the same place}
R = {(x, y): x and y work at the same place}
Suppose we have four boxes A, B, C, D containing coloured marbles as given below:
One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from,
One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from,
- Box A?
- Box B?
- Box C?