Evaluate the following integrals: ^ x ( 4x-4 1- 4x )dx - Vidyadip

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
$=\text{e}^{\text{x}}\cot2\text{x}-\int\text{e}^{\text{x}}\frac{\text{d}}{\text{dx}}(\cot2\text{x})\text{dx}-2\int\text{e}^{\text{x}}\text{cosec}^22\text{x dx}$
$=\text{e}^{\text{x}}\cot2\text{x}+2\int\text{e}^{\text{x}}\text{cosec}^22\text{x}-2\int\text{e}^{\text{x}}\text{cosec}^22\text{x dx}$
$=\text{e}^{\text{x}}\cot2\text{x}+\text{C}$

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

Using integeration, find the area of the region bounded by the y - 1 = x, the x-axis and the ordinates x = -2 and x = 3.

Evaluate the following integrals:
$\int_{0}^\limits{1}\frac{24\text{x}^3}{(1+\text{x}^2)^4}\text{ dx}$

In a quadrilateral $ABCD,$ prove that $AB^2+ BC^2+ CD^2+ DA^2= AC^2+ BD^2+ 4PQ^2,$ where $P$ and $Q$ are middle points of diagonals $AC$ and $BD.$

If $xy = e^{(x–y)}$, then show that $\frac{\text{dy}}{\text{dx}} = \frac{\text{y (x - 1)}}{\text{x (y + 1)}}.$

$\text{If}\ \text{y}=\sin(\sin\text{x}),\ \text{prove that}\frac{\text{d}^2\text{y}}{\text{dx}^2}+\tan\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}\cos^2\text{x}=0.$

Solve the following differential equation:
$3\text{x}^2\text{dy}=(3\text{xy}+\text{y}^2)\text{dx}$

If a variable line in two adjacent positions has direction cosines l, m, n and $\text{l}+\delta\text{l},\text{m}+\delta\text{m},\text{n}+\delta\text{n},$ show that the small angle $\delta\theta$ between the two positions is given by $\delta\theta^2=\delta\text{l}^2+\delta\text{m}^2+\delta\text{n}^2.$

Differentiate the following functions with respect to x:
$\Big(\text{x}+\frac{1}{\text{x}}\Big)^\text{x}+\text{x}^{\Big(1+\frac{1}{\text{x}}\Big)}$

Consider $\text{f : R} - \left\{-\frac{4}{3}\right\} \rightarrow \text{R} - \left\{\frac{4}{3}\right\} \text{given by f(x)} = \frac{\text{4x + 3}}{\text{3x + 4}}.$ Show that f is bijective. Find the inverse of f and hence find $f ^{–1}(0)$ and x such that $f ^{–1}(x) = 2$.

Using differentials, find the approximate values of the following:
$\log_\text{e}10.02$ it being given that $\log_\text{e}10=2.3026$