Evaluate the following integrals: x (x^2+2x+2) x+1 dx - Vidyadip

Question

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

✓

Answer

Let $\text{I}=\int\frac{\text{x}}{(\text{x}^2+2\text{x}+2)\sqrt{\text{x}+1}}\text{ dx}$
Let $\text{x}+1=\text{t}^2$
$\text{dx}=2\text{t dt}$
$=2\int\frac{(\text{t}^2-1)\text{t dt}}{(\text{t}^4+1)\text{t}}$
$=2\int\frac{(\text{t}^2-1)\text{ dt}}{(\text{t}^4+1)}$
$=2\int\frac{\big(1-\frac{1}{\text{t}^2}\big)\text{ dt}}{\text{t}+\frac{1}{\text{t}^2}}$
$=2\int\frac{\big(1-\frac{1}{\text{t}^2}\big)}{\big(\text{t}+\frac{1}{\text{t}}\big)^2-2}$
Let $\text{t}+\frac{1}{\text{t}}=\text{y}$
$\Big(1-\frac{1}{\text{t}^2}\Big)\text{ dt}=\text{dy}$
$\therefore\ \text{I}=2\int\frac{\text{dy}}{\text{y}^2-2}$
$=\frac{2}{2\sqrt{2}}\log\bigg|\frac{\text{y}-\sqrt{2}}{\text{y}+\sqrt{2}}\bigg|+\text{C}$
Thus, $\text{I}=\frac{1}{\sqrt{2}}\log\bigg|\frac{\text{t}^2+1-\sqrt{2}\text{t}}{\text{t}^2+1+\sqrt{2}\text{t}}\bigg|+\text{C}$
Hence,
$\text{I}=\frac{1}{\sqrt{2}}\log\begin{vmatrix}\frac{\text{x}+2-\sqrt{2(\text{x}+1)}}{\text{x}+2\sqrt{2(\text{x}+1)}}\end{vmatrix}+\text{C}$

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 "5 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

Evaluate the following integrals:$\int\frac{\sin2\text{x}}{\sqrt{\sin^4\text{x}+4\sin^2\text{x}-2}}\text{ dx}$

If $\text{y}=\cos^{-1}(2\text{x})+2\cos^{-1}\sqrt{1-4\text{x}^2}, -\frac{1}{2}<\text{x}<0,$ find $\frac{\text{dy}}{\text{dx}}.$

A man 180cm tall walks at a rate of 2m/ sec. away, from a source of light that is 9m above the ground. How fast is the length of his shadow increasing when he is 3m away from the base of light?

Find the vector equation of the following planes in non-parametric form.
$\vec{\text{r}}=(\lambda-2\mu)\hat{\text{i}}+(3-\mu)\hat{\text{j}}+(2\lambda+\mu)\hat{\text{k}}$

Find the distance of the point (3, 3, 3) from the plane $\vec{\text{r}}\cdot(5\hat{\text{i}}+2\hat{\text{j}}-7\hat{\text{k}})+9=0$

Without expanding, prove that:
$\begin{vmatrix}\text{a}&\text{b}&\text{c}\\\text{x}&\text{y}&\text{z}\\\text{p}&\text{q}&\text{r}\end{vmatrix}=\begin{vmatrix}\text{x}&\text{y}&\text{z}\\\text{p}&\text{q}&\text{r}\\\text{a}&\text{b}&\text{c}\end{vmatrix}=\begin{vmatrix}\text{y}&\text{b}&\text{q}\\\text{x}&\text{a}&\text{p}\\\text{z}&\text{c}&\text{r}\end{vmatrix}$

Find the points of discontinuity, if any of the following function:
$\text{f(x)}=\begin{cases}\text{x}^3-\text{x}^2+2\text{x}-2,&\text{if }\text{ x}\neq1\\4,&\text{if }\text{ x}=1\end{cases}$

Find the shortest distance between the lines
$\vec{\text{r}}=\big(\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}\big)+\lambda\big(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\big)$ and, $\vec{\text{r}}=2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}+\mu\big(2\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}\big)$

If A and B are independent events such that P(A) = p, P(B) = 2p and P(Exactly one of A and B occurs) $=\frac{5}{9},$ then find the value or p.

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}+\sin\Big(\frac{\text{y}}{\text{x}}\Big)$