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

Question

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

✓

Answer

$\text{I} =\frac{1}{2} \int \frac{\text{1}}{\text{(2 + t) (4 +} \text{t}^{2})}\text{dt} \text{ } \text{where} \ \text{t} = \text{x}^2$
$\text{Now},\Bigg[ \frac{1}{\text{(2 + t) (4 + t}^{2})}\Bigg] =\frac{1}{2}\Bigg[ \frac{1}{8 (2 + \text{t})} - \frac{1}{8} \bigg(\frac{\text{t - 2}}{\text{4 + t}^{2}}\bigg)\Bigg]$
$\Rightarrow \int \frac{\text{1}}{\text{(2 + t) (4 + t}^{2})} \text{dt}= \frac{1}{16} \log | \text{2 + t|} = -\frac{1}{32} \log |\text{4 + t}^{2}| + \frac{1}{16} \tan^{-1} \bigg(\frac{\text{t}}{2}\bigg) + \text{c}$
$\Rightarrow \int \frac{\text{x}}{(2 + \text{x}^{\text{2}}) (4 + \text{x}^{4\text{}})}\text{dx} = \frac{1}{16} \log |\text{2 + x}^{\text{2}}| - \frac{1}{32} \log \text{|4 + x}^{\text{4}}| + \frac{1}{16} \tan^{-1} \bigg(\frac{\text{x}^{\text{2}}}{2}\bigg) + \text{c}$

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 "5 Marks Questions" from INTEGRALS→INTEGRALS — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Minimise and Maximise Z = x + 2y
subject to $\text{x}+2\text{y}\geq100,\ 2\text{x}-\text{y}\leq0,\ 2\text{x}+ \text{y}\leq200;\ \text{x},\ \text{y}\geq0.$

If $\text{A}=\begin{bmatrix}1&2\\2&1\end{bmatrix}, f(x) = x^2 - 2x - 3, $ show that $f(A) = 0$

If $\text{y}\sqrt{\text{x}^2+1}=\log\Big(\sqrt{\text{x}^2+1}-\text{x}\Big),$ prove that $\big(\text{x}^2+1\big)\frac{\text{dx}}{\text{dx}}+\text{xy}+1=0$

Using properties of determinants, prove that:
$\begin{vmatrix}\text{x}&\text{x}^2&1+\text{px}^3\\\text{y}&\text{y}^2&1+\text{py}^3\\\text{z}&\text{z}^2&1+\text{pz}^3\end{vmatrix}=(1+\text{pxyz})(\text{x}-\text{y})(\text{y}-\text{z})(\text{z}-\text{x}),$ where $p$ is any scalar.

Solve the differential equation $(1 + y^2) \tan^{-1}x dx + 2y (1 + x^2) dy = 0.$

Sketch the graph y = |x - 5|. Evaluate $\int\limits_{0}^{1}|\text{x}-5|\text{dx} $ . What does this value of the integral represent on the graph.

Evaluate the following integrals:
$\int_{1}^\limits{2}\frac{1}{\text{x}\big(1+\log\text{x}\big)^2}\text{ dx}$

A bag contains 4 white, 7 black and 5 red balls. Three balls are drawn one after the other without replacement. Find the probability that the balls drawn are white, black and red respectively.

Find the probability that the sum of the numbers showing on two dice is 8, given that at least one die does not show five.

Evaluate the following integrals:
$\int_{0}^\limits{\frac{\pi}{2}}\frac{\text{dx}}{\text{a}\cos\text{x}+\text{b}\sin\text{x}}\text{ a},\text{b}>0$