Find: 2x + 1 (x^ 2 + 1) (x^ 2 + 4) dx - Vidyadip

Question

Find: $\int\frac{2x + 1}{(x^{2} + 1) (x^{2} + 4)}dx$

✓

Answer

$\text{Let I} = \int\frac{\text{2x} + 1}{(\text{x}^{2} + 1) (\text{x}^{2} + 4)}\text{dx}$
$\text{Let}\frac{\text{2x} + 1}{(\text{x}^{2} + 1) (\text{x}^{2} + 4)} = \frac{\text{Ax + B}}{\text{x}^{2} + 1} + \frac{\text{Cx + D}}{\text{x}^{2} + 4}$
$\text{Getting A} = \frac{2}{3}, \text{B} = \frac{1}{3}, \text{C} = \frac{-2}{3} ,\text{D} = \frac{-1}{3}$
$\therefore\text{I} = \frac{2}{3}\int\frac{\text{x}}{\text{x}^{2} + 1}\text{dx}+\frac{1}{3}\int\frac{\text{1}}{\text{x}^{2} + 1}\text{dx}+ \frac{-2}{3}\int\frac{\text{xdx}}{\text{x}^{2} + 4}\text{dx} + \frac{-1}{3}\int\frac{\text{dx}}{\text{x}^{2} + 4}$
$= \frac{1}{3}\log|\text{x}^{2} + 1| + \frac{1}{3} \tan^{-1} \text{x} -\frac{1}{3}\log|\text{x}^{2} + 4| -\frac{1}{6}\tan^{-1}\frac{\text{x}}{2} + \text{C}$

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 "5 Marks Questions" from APPLICATION OF DERIVATIVES→APPLICATION OF DERIVATIVES — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the equation of the plane passing through the point (-1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.

If $\text{P}=\begin{bmatrix}\text{x}&0&0\\0&\text{y}&0\\0&0&\text{z}\end{bmatrix}$ and $\text{Q}=\begin{bmatrix}\text{a}&0&0\\0&\text{b}&0\\0&0&\text{c}\end{bmatrix},$ prove that $\text{PQ}=\begin{bmatrix}\text{xa}&0&0\\0&\text{y}\text{b}&0\\0&0&\text{zc}\end{bmatrix}=\text{QP}$

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{x}=\text{a}\sec\text{t},\text{y}=\text{b}\tan\text{t}\text{ at }\text{t}$

Evaluate $\int_{-1}^1 5 x^4 \sqrt{x^5+1} d x$.

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}+1=\text{e}^{\text{x + y}}$

Find the matrix A such that
$\text{A}=\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}=\begin{bmatrix}-7&-8&-9\\2&4&6\end{bmatrix}$

If $\vec{\text{a}}=2\hat{\text{i}}-3\hat{\text{j}}+\hat{\text{k}},\vec{\text{b}}=-\hat{\text{i}}+\hat{\text{k}},\vec{\text{c}}=2\hat{\text{j}}-\hat{\text{k}}$are three vectors, find the aera of the parallelogram having diagonals $\big(\vec{\text{a}}+\vec{\text{b}}\big)$ and $\big(\vec{\text{b}}+\vec{\text{c}}\big).$

Find the intervals in which the function f given by
f(x) = sin x + cos x, 0 < x < 2 $\pi$.
is strictly increasing or strictly decreasing.

Examine the applicability of Mean Value Theorem for all three functions given in the above exercise 2.
$\text{f(x)}=[\text{x}]\text{ for x}\in[-2,\ 2]$

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{3}}_{\frac{\pi}{6}}\frac{\sqrt{\sin\text{x}}}{\sqrt{\sin\text{x}}+\sqrt{\cos\text{x}}}\text{ dx}$