Evaluate the following definite integrals: _ 1 ^ 2 x (x+1)(x+2) d… - Vidyadip

Question

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

✓

Answer

Let $\text{I}=\int_{1}^\limits{2}\frac{\text{x}}{(\text{x}+1)(\text{x}+2)}\text{ dx}$ Then,
$\text{I}=\int_{1}^\limits{2}\Big(\frac{-1}{(\text{x}+1)}+\frac{2}{(\text{x}+2)}\Big)\text{dx}$
$\Rightarrow\text{I}=-\int_{1}^\limits{2}\frac{1}{(\text{x}+1)}\text{ dx}+2\int_{1}^\limits{2}\frac{1}{(\text{x}+2)}\text{ dx}$
$\Rightarrow\text{I}=\big[-\log(\text{x}+1)+2\log(\text{x}+2)\big]^2_1$
$\Rightarrow\text{I}=-\log3+2\log4+\log2-2\log3$
$\Rightarrow\text{I}=5\log2-3\log3$
$\Rightarrow\text{I}=\log2^5-\log3^3$
$\Rightarrow\text{I}=\log\frac{32}{27}$

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 Definite Integrals→Definite 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 differential equation
$\frac{\text{dy}}{\text{dx}}=\cos^3\text{x}\sin^2\text{x}+\text{x}\sqrt{2\text{x}+1}$

Differentiate $\tan^{-1}\Big(\frac{\cos\text{x}}{1+\sin\text{x}}\Big)$ with respect to $\sec^{-1}\text{x}$

Solve the following differential equations:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{x + y}}{\text{x}-\text{y}}$

If $\text{f}\text{(x)}=\begin{cases}\frac{\text{x}^2-1}{\text{x}-1},& \text{for }\text{ x}\neq1 \\2,&\text{for }\text{ x}=1\end{cases}$ Find whether f (x) is continuous at x = 1.

Evaluate the following integrals:
$\int\limits^{1}_0\frac{\log(1+\text{x})}{1+\text{x}^2}\text{ dx}$

If $\text{f}\text{(x)}=\begin{cases}\frac{1-\cos\text{kx}}{\text{x}\sin\text{x}}, & \text{x} \neq 0\\\frac{1}{2}, & \text{x}= 0\end{cases}$ is continuous at x = 0. find k.

Find $\frac{d y}{d x}$
$(a) \ \sqrt{x^2+y^2}=\log _e\left(x^2-y^2\right)$
$(b) \ y=x^{x^{x^x} \ldots......\infty}$
$(c) \ x y \log (x+y)=1$

Determine the values of a, b, c for which the function
$\text{f}\text{(x)}=\begin{cases}\frac{\sin\text{(a}+1)\text{x}+\sin\text{x}}{\text{x}}, &\text{for}\text{ x}<0,&\\\text{ c},&\text{for x}=0\\\frac{\sqrt{\text{x}+\text{bx}^2}-\sqrt{\text{x}}}{\text{bx}^\frac{3}{2}},&\text{for x}>0\end{cases}$ is continuous at x = 0.

Find a point on the curve $y = x^3 + 1$ where the tangent is parallel to the chord joining (1, 2) and (3, 28).

If the radius of a sphere is measured as 9cm with an error of 0.03m, find the approximate error in calculating its surface area.