Evalute : x^5 x^2+1 d x - Vidyadip

Question

Evalute : $\int \frac{x^5}{x^2+1} d x$

✓

Answer

Let $I=\int \frac{x^5}{x^2+1} d x=\int \frac{x^4}{x^2+1} \cdot x d x$
$
=\int \frac{\left(x^2\right)^2}{x^2+1} \cdot x d x
$
Put $x^2+1=t \quad \therefore 2 x d x=d t$
$
\begin{aligned}
& \therefore x d x=\frac{d t}{2} \text { and } x^2=t-1 \\
& \begin{aligned}
\therefore I & =\int \frac{(t-1)^2}{t} \cdot \frac{d t}{2}=\frac{1}{2} \int\left(\frac{t^2-2 t+1}{t}\right) d t \\
& =\frac{1}{2} \int\left(t-2+\frac{1}{t}\right) d t \\
& =\frac{1}{2} \int t d t-\int 1 d t+\frac{1}{2} \int \frac{1}{t} d t \\
& =\frac{1}{2} \cdot \frac{t^2}{2}-t+\frac{1}{2} \log |t|+c
\end{aligned}
\end{aligned}
$
$
=\frac{1}{4}\left(x^2+1\right)^2-\left(x^2+1\right)+\frac{1}{2} \log \left|x^2+1\right|+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 "Solve the Following Question.(3 Marks)" from Integration (p-1)→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

Find $\frac{d y}{d x}$ if, :
$
y=\log \left(10 x^4+5 x^3-3 x^2+2\right)
$

$\int_0^1 \frac{x^2+3 x+2}{\sqrt{x}} d x$

If the difference between true discount and banker’s discount on a sum due 4 months hence is $\text{₹} 20.$ Find true discount, banker’s discount and amount of bill, the rate of simple interest charged is $5\%$ p.a.

4,000 bedsheets worth ₹ 6,40,000 were insured for $\left(\frac{3}{7}\right)^{t h}$ of their value. Some of the bedsheets were damaged in the rainy season and were reduced to $40 \%$ of their value. If the amount recovered against damage was ₹ 32,000. Find the number of damaged bedsheets.

Solve the following problem :

Solve the following assignment problem to maximize sales:

Salesman	Territories
Salesman	I	II	III	IV	V
A	11	16	18	15	15
B	7	19	11	13	17
C	9	6	14	14	7
D	13	12	17	11	13

Find expected value and variance of X, the number on the uppermost face of a fair die.

Heramb requires at most 400 calories from his breakfast. Every morning he likes to take oats and milk. If each bowl of oats and a glass of milk provides him 80 calories and 50 calories respectively, then as a constraint this information can be expressed as ______

In a partially destroyed laboratory record of an analysis of regression data, the following data are legible:
Variance of X = 9
Regression equations:
8x – 10y + 66 = 0 And 40x – 18y = 214.
Find on the basis of the above information
(i) The mean values of X and Y.
(ii) Correlation coefficient between X and Y.
(iii) Standard deviation of Y.

If the feasible region is bounded by the inequations 2x + 3y ≤ 12, 2x + y ≤ 8, 0 ≤ x, 0 ≤ y, then point (5, 4) is a ______ of the feasible region

Divide the number $20$ into two parts such that their product is maximum