Integrate the function in Exercise: e^ 3 (x^ 4 +1 )^ -1 - Vidyadip

Question

Integrate the function in Exercise:

$\text{e}^{3}\log\text{x}\big(\text{x}^{4}+1\big)^{-1}$

✓

Answer

$ \text{e}^{3\log\text{x}}.\big(\text{x}^{4}+1\big)^{-1}=\text{e}^{\log\text{x}^{3}}.\big(\text{x}^{4}+1\big)^{-1}=\frac{\text{x}^{3}}{\big(\text{x}^{4}+1\big)}$
$\text{Let}\ \text{x}^{4}+1=\text{t}\Rightarrow 4\text{x}^3\ \text{dx}=\text{dt}$
$\Rightarrow\int\text{e}^{3\log\text{x}}\big(\text{x}^{4}+1\big)^{-1}\text{dx}=\int\frac{\text{x}^{3}}{\big(\text{x}^{4}+1\big)}\text{dx}$
$=\frac{1}{4}\int\frac{\text{dt}}{\text{t}}$
$=\frac{1}{4}\log|\text{t}|+\text{C}$
$=\frac{1}{4}\log\big(\text{x}^{4}+1\big)+\text{C}$

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 "2 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

Find the value of $\int \frac{1}{x-\sqrt{x}} d x$.

Write the primitive or anti-derivative of $\text{f(x)}=\sqrt{\text{x}}=\frac{1}{\sqrt{\text{x}}}$.

Evaluate the following integrals:
$\int\limits^3_2\frac{1}{\text{x}}\text{ dx}$

Find $\frac{\text{dy}}{\text{dx}},$ when
$\text{x}=\text{a}(1-\cos\theta)\text{ and y}=\text{a}(\theta+\sin\theta)\text{ at }\theta=\frac{\pi}{1}$

Find the components along the coordinate axis of the position vector of the following point:
P(3, 2)

If $\vec{\text{a}}$ is a unit vector, then find $|\vec{\text{x}}|$ in each of the following.

$\big(\vec{\text{x}}-\vec{\text{a}}\big).\big(\vec{\text{x}}+\vec{\text{a}}\big)=8$

Integrate the rational function in exercise:
$\frac{1}{\text{x}^2-9}$

Write the value $\big(\hat{\text{i}}\times\hat{\text{j}}\big).\hat{\text{k}}+\hat{\text{i}}.\hat{\text{j}}$

Fatima and John appear in an interview for two vacancies for the same post. The probability of Fatima's selection is $\frac{1}{7}$ and that of John's selection is $\frac{1}{5}$. What is the probability that,
Only one of them will be selected?

For the matrix $A=\left[\begin{array}{ll} {1} & {5} \\ {6} & {7} \end{array}\right]$, verify that (A + A′) is a symmetric matrix.