Evaluate the following definite integrals : _0^1 x^2+3 x+2 x d x - Vidyadip

Question

Evaluate the following definite integrals : $\int_0^1 \frac{x^2+3 x+2}{\sqrt{x}} d x$

✓

Answer

$
\begin{aligned}
& \int_0^1 \frac{x^2+3 x+2}{\sqrt{x}} d x \\
= & \int_0^1\left(\frac{x^2}{\sqrt{x}}+\frac{3 x}{\sqrt{x}}+\frac{2}{\sqrt{x}}\right) d x \\
= & \int_0^1\left(x^{\frac{3}{2}}+3 x^{\frac{1}{2}}+2 x^{-\frac{1}{2}}\right) d x \\
= & {\left[\frac{x^{\frac{5}{2}}}{5 / 2}+3\left(\frac{x^{\frac{3}{2}}}{3 / 2}\right)+2\left(\frac{x^{\frac{1}{2}}}{1 / 2}\right)\right]_0^1 } \\
= & {\left[\frac{2}{5} x^{\frac{5}{2}}+2 x^{\frac{3}{2}}+4 x^{\frac{1}{2}}\right]_0^1 } \\
= & {\left[\frac{2}{5}(1)^{\frac{5}{2}}+2(1)^{\frac{3}{2}}+4(1)^{\frac{1}{2}}\right]-(0+0+0) } \\
= & \frac{2}{5}+2+4=\frac{32}{5} .
\end{aligned}
$

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 "Solve the Following Question.(2 Marks)" from Definite Integration (p-1)→Definite Integration (p-1) — all question types→Maths (commerce) STD 12 Commerce / Arts question papers→Maharashtra Board STD 12 Commerce / Arts question papers→Maharashtra Board question paper generator→

Similar questions

Construct the matrix $A=\left[a_{i j}\right]_{3 \times 3}$, where $a_{i j}=i-j$. State whether $A$ is symmetric or skew-symmetric.

A Company manufactures two types of fertilizers $F_1$_ and $F_2$. Each type of fertilizer requires two raw materials A and B. The number of units of A and B required to manufacture one unit of fertilizer $F_1$_ and $F_2$ and availability of the raw materials A and B per day are given in the table below

	$F_1$	$F_2$	Availability
A	$2$	$3$	$70$
B	$1$	$4$	$70$

By selling one unit of $F_1$ and one unit of $F_2$, the company gets a profit of $₹ 500 $and $₹ 750$ respectively. Formulate the problem as LPP to maximize the profit.

Using truth table prove that p ˅ (q ˄ r) ≡ (p ˅ q) ˄ (p ˅ r).

Evalute : $\int x^3 e^{x^2} d x$

Find $\frac{d y}{d x}$ if $x =5 t ^2, y =10 t$

Find the rate of change of demand (x) of a commodity with respect to price (y) if :
$
y=12+10 x+25 x^2
$

Prepare the truth tables for the following statement patterns : p → (~p ∨ q)

If $\sum p_0q_0 = 120, \sum p_0q_1 = 160, \sum p_1q_1 = 140$, and $\sum p_1q_0 = 200$ find Laspeyre’s, Paasche’s, Dorbish-Bowley’s, and Marshall-Edgeworth’s Price Index Numbers.

Calculate the cost of living index.

Group	Food	Clothing	Fuel & Lighting	House Rent	Miscellaneous
I	70	90	100	60	80
W	5	3	2	4	6

Check whether following matrices are invertible or not: $\left[\begin{array}{lll}3 & 4 & 3 \\ 1 & 1 & 0 \\ 1 & 4 & 5\end{array}\right]$