Find: e^ x (2 + e^ x ) (4 + e^ 2x ) dx. - Vidyadip

Question

Find:
$\int \frac{\text{e}^{\text{x}}}{(2 + \text{e}^{\text{x}}) (4 + \text{e}^{2\text{x}})} \text{dx}.$

✓

Answer

$\text{I} = \int \frac{\text{dt}}{\text{(2 + t) (4 +} \text{t}^{2})} \text{ } \text{where} \text{ e}^{\text{x}} = \text{t}$
$\text{Now}, \frac{1}{\text{(2 + t) (4 + t}^{2})} = \frac{1}{8 (2 + \text{t})} - \frac{1}{8} \bigg(\frac{\text{t - 2}}{\text{4 + t}^{2}}\bigg)$
$\Rightarrow \int \frac{\text{dt}}{\text{(2 + t) (4 + t}^{2})} = \frac{1}{8} \log | \text{2 + t|} = \frac{1}{16} \log |\text{4 + t}^{2}| + \frac{1}{8} \tan^{-1} \bigg(\frac{\text{t}}{2}\bigg) + \text{c}$
$\Rightarrow \int \frac{\text{e}^{\text{x}}\text{dx}}{(2 + \text{e}^{\text{x}}) (4 + \text{e}^{2\text{x}})} = \frac{1}{8} \log |\text{2 + e}^{\text{x}}| - \frac{1}{16} \log \text{|4 + e}^{\text{2x}}| + \frac{1}{8} \tan^{-1} \bigg(\frac{\text{e}^{\text{x}}}{2}\bigg) + \text{c}$

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

The population of a village increase continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20,000 in 1999 and 25,000 in the year 2004, what will be the population of the village in 2009?

Evaluate the following integrals:
$\int\limits^{\text{a}}_{-\text{a}}\sqrt{\frac{\text{a}-\text{x}}{\text{a}+\text{x}}}\text{ dx}$

Differentiate the following functions with respect to x:
$\text{x}^{\sin{\text{x}}}$

Using the method of interation, find the area of the region bounded by the following lines:
3x - y - 3 = 0, 2x + y - 12 = 0, x - 2y - 1 = 0.

By using the properties of definite integral, evaluate the integral $\int_0^{\frac{\pi}{2}}(2 \log \sin x-\log \sin 2 x) d x$

Solve the following differential equations:
$\frac{\text{dy}}{\text{dx}}=2\text{e}^{\text{x}}\text{y}^3,\text{y}(0)=\frac{1}{2}$

Let S be the set of all real numbers except -1 and let '*' be an operation defined by a * b = a + b + ab for all a, b ∈ S. Determine whether '*' is a binary operation on S. If yes, check its commutativity and associativity. Also, solve the equation (2 * x) * 3 = 7.

A company sells two different products, A and B. The two products are produced in a common production process, which has a total capacity of 500 man-hours. It takes 5 hours to produce a unit of A and 3 hours to produce a unit of B. The market has been surveyed and company officials feel that the maximum number of unit of A that can be sold is 70 and that for B is 125. If the profit is Rs. 20 per unit for the product A and Rs. 15 per unit for the product B, how many units of each product should be sold to maximize profit?

Determine the area under the cutve $\text{y}=\sqrt{\text{x}^{2}-\text{x}^{2}}$ included between the lines x = 0 and x = a.

Evaluate the following intregals:
$\int\frac{\text{x}}{\sqrt{\text{x}^2+6\text{x}+10}}\ \text{dx}$