1 x^ 1 3 (x^ 1 3 -1 ) dx - Vidyadip

Question

$\int\frac{1}{\text{x}^{\frac{1}{3}}\big(\text{x}^{\frac{1}{3}}-1\big)}\text{dx}$

✓

Answer

Let $\text{I}=\int\frac{1}{\text{x}^{\frac{1}{3}}\big(\text{x}^{\frac{1}{3}}-1\big)}\text{dx}$
$=\int\frac{1}{\text{x}^{\frac{2}{3}}-\text{x}^{\frac{1}{3}}}\text{dx}$
Let $\text{x}=\text{t}^{3}$
On differentiating both sides, we get
$\text{dx}=3\text{t}^{2}\text{dt}$
$\therefore\ \text{I}\int\frac{3\text{t}^{2}}{(\text{t})^{\frac{2}{3}}-(\text{t}^{3})^{\frac{1}{3}}}\text{dt}$
$=\int\frac{3\text{t}^{2}}{\text{t}^2-\text{t}}\text{dt}$
$=3\int\frac{\text{t}}{\text{t}-1}\text{dt}$
$=3\int\frac{(\text{t}-1)+1}{\text{t}-1}\text{dt}$
$=3\int\Big[(1)+\frac{1}{\text{t}-1}\Big]\text{dt}$
$=\big[1+\log(\text{t}-1)\big]+\text{C}$
$=3\text{x}^\frac{1}{3}+3\log\big({\text{x}^\frac{1}{3}-1\big)}+\text{C}$
Hence, $\int\frac{1}{\text{x}^{\frac{1}{3}}\big(\text{x}^{\frac{1}{3}}-1\big)}\text{dx}=3\text{x}^\frac{1}{3}+3\log\big({\text{x}^\frac{1}{3}-1\big)}+\text{C}$

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 "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

Solve the following equation for x:
$\tan^{-1}(\text{x}+2)+\tan^{-1}(\text{x}-2)=\tan^{-1}\Big(\frac{8}{79}\Big),\text{x}>0$

Let R be the relation defined on the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b): both a and b are either odd or even}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each other and all the elements of the subset {2, 4, 6} are related to each other, but no element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}.

Find a unit vector perpendicular to each of the vectors $\vec{\text{a}}+\vec{\text{b}}$ and $\vec{\text{a}}-\vec{\text{b}},$ where $\vec{\text{a}}=3\hat{\text{i}}+2\hat{\text{j}}+2\hat{\text{k}}$ and $\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-2\hat{\text{k}}.$

Determine the maximum distance that the man can travel.

Solve the following differential equation:
$\cos^{2}\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=\tan\text{x}.$

$\text{Find} \int \frac{\sqrt{x}}{\sqrt{\text{a}^{3}} - \text{x}^{3}}\text{dx}.$

Evaluate the following intregals:
$\int\frac{\text{x}^2+\text{x}+1}{(\text{x}^2+1)(\text{x}+2)}\text{ dx}$

There are two types of fertilizers F₁and F₂. F₁consists of 10% nitrogen and 6% phosphoric acid and F₂consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds the she needs atleast 14kg of nitrogen and 14kg of phosphoric acid for her crop. If F₁costs Rs 6/kg and F₂costs Rs 5/kg, determine how much of each type of fertilizer should be used so that the nutrient requirements are met at minimum cost. What is the minimum cost?

Find the vector equation of the plane passing through three point with position vectors $\hat{\text{i}}+\hat{\text{j}}-2\hat{\text{k}},2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$ and $\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}.$ Also, find coordinates of the point of intersection of this plane and the line $\vec{\text{r}}=3\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}+\lambda(2\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}).$

Show that the semi-vertical angle of the cone of the maximum volume and of given slant height $(l)$ is $\tan ^{-1} \sqrt{2}$.