Evaluate (1-x) x dx - Vidyadip

Question

Evaluate $\int(1-\text{x})\sqrt{\text{x}}\text{ dx}$

✓

Answer

Let $\sqrt{\text{x}}=\text{t}$
$\text{x}=\text{t}^2$
$1-\text{x}=1-\text{t}^2$
$-\text{dx}=-2\text{tdt}$
$\text{dx}=2\text{tdt}$
$\text{I}=\int(1-\text{x})\sqrt{\text{x}}\text{ dx}$
$=\int(1-\text{t}^2)\text{t}2\text{t dt}$
$=2\int (1-\text{t}^2)\text{t}^2\text{ dt}$
$=2\big(\int\text{t}^2\text{dt}-\int\text{t}^4\text{dt}\big)$
$=2\frac{\text{t}^3}{3}-2\frac{\text{t}^5}{5}+\text{C}$
$=\frac{2}{3}\text{x}^{\frac{3}{2}}-\frac{2}{5}\text{x}^{\frac{5}{2}}+\text{C}$

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 Questions" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Integrate the function: $x \sqrt{x+2}$

If $\begin{bmatrix}1&0\\\text{y}&5\end{bmatrix}+2\begin{bmatrix}\text{x}&0\\1&-2\end{bmatrix}=\text{I},$ where I is 2×2 unit matrix. Find x and y.

Construct a $2 \times 3$ matrix $A = [a_{ij}]$ whose elements $a_{ij}$ are give by:$\text{a}_\text{ij}=\frac{(\text{i}+\text{j})^2}{2}$

A line passes through the point with position vector $2\hat{\text{i}} - 3\hat{\text{j}} + 4\hat{\text{k}}$ and is perpendicular to the plane $\vec{\text{r}}. (3\hat{\text{i}} + 4\hat{\text{j}} - 5\hat{\text{k}}) = 7.$ Find the equation of the line in cartesian and vector forms.

Form the differential equation from the following primitives where constants are arbitrart:$\text{y}=\text{cx}+2\text{c}^2+\text{c}$

The random variable X has a probability distribution P(X) of the following form, where ‘k’ is some number.
$\text{P}(\text{X}=\text{x})=\begin{cases}\text{k}, & \text{if x}=0\\2\text{k}, & \text{if x}=1\\3\text{k}, & \text{if x}=2\\0, & \text{otherwise}\end{cases}$
Determine the value of ‘k’.

Let A and B be square matrices of the same order. Does $(A + B)^2 = A^2 + 2AB + B^2$ hold? If not, why?

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

Find the area of the region bounded by the line y = 3x + 2, the x-axis and the ordinates x = - 1 and x = 1

The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves ₹ 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?