Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluate the following integrals:
$\int(2\text{x}+3)\sqrt{\text{x}^2+4\text{x}+3}\text{dx}$
✓

Answer

Let $\text{I}=\int(2\text{x}+3)\sqrt{\text{x}^2+4\text{x}+3}\text{dx}$Let $(2\text{x}+3)=\lambda\frac{\text{d}}{\text{dx}}(\text{x}^2+4\text{x}+3)+\mu$
$=\lambda(2\text{x}+4)+\mu$ Equating similar terms, we get, $\lambda=1\text{ and }4\lambda+\mu=3$ $\Rightarrow\mu=-1$ So, $\text{I}=\int((2\text{x}+4)+(-1))\sqrt{\text{x}^2+4\text{x}+3}\text{dx}$ $=\int(2\text{x}+4)\sqrt{\text{x}^2+4\text{x}+3}\text{dx}-\int\sqrt{\text{x}^2+4\text{x}+3}\text{dx}$ Let $\text{x}^2+4\text{x}+3=\text{t}$ $\Rightarrow(2\text{x}+4)\text{dx = dt}$ $\therefore\ \text{I}=\int\sqrt{\text{t}}\text{dt}-\int\sqrt{(\text{x}+2)^2-1}\text{dx}$ $=\frac{3}{2}\text{t}^{\frac{3}{2}}-\frac{(\text{x}+2)}{2}\sqrt{\text{x}^2+4\text{x}+3}\\+\frac{1}{2}\log\Big|(\text{x}+2)+\sqrt{\text{x}^2+4\text{x}+3}\Big|+\text{C}$ Hence, $\text{I}=\frac{2}{3}(\text{x}^2+4\text{x}+3)^{\frac{3}{2}}-\Big(\frac{\text{x}+2}{2}\Big)\sqrt{\text{x}^2+4\text{x}+3}\\+\frac{1}{2}\log\Big|(\text{x}+2)+\sqrt{\text{x}^2+4\text{x}+3}\Big|+\text{C}$

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

If the derivative of $\tan^{-1} (a + bx)$ takes the value $1$ at $x = 0$, prove that $1 + a^2 = b$.
→
Find the shortest distance between the lines $\vec{r}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(\hat{i}-3 \hat{j}+2 \hat{k})$ and $\vec{r}=(4 \hat{i}+5 \hat{j}+6 \hat{k})+\mu(2 \hat{i}+3 \hat{j}+\hat{k})$
→
If $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ are non-coplanar vectors, prove that the point having the following position vectors is collinear:$\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}},\ 4\vec{\text{a}}+3\vec{\text{b}},\ 10\vec{\text{a}}+7\vec{\text{b}}-2\vec{\text{c}}$
→
Differentiate the following functions with respect to x:
$\sin^{-1}\Big\{\frac{\sqrt{1+\text{x}}+\sqrt{1-\text{x}}}{2}\Big\},0<\text{x}<1$
→
Evaluate the following integrals:
$\int\limits^{\pi}_0\sin^3\text{x}(1+2\cos\text{x})(1+\cos\text{x})^2\text{ dx}$
→
Evaluate the following integrals:$\int\frac{2\text{x}-3}{\text{x}^2+6\text{x}+13}\text{ dx}$
→
AB is a diameter of a circle and C is any point on the circle. Show that the area of $\triangle\text{ABC}$ is maximum, when it is isosceles.
→
Let R be a relation on the set N given by R = {(a, b): a = b - 2, b > 6}. Then,
  1. (2, 4) ∈ R
  2. (3, 8) ∈ R
  3. (6, 8) ∈ R
  4. (8, 7) ∈ R
→
If $\text{A}=\begin{bmatrix}-2\\4\\5\end{bmatrix},\text{B}=\begin{bmatrix}1&3&-6\end{bmatrix},$ verify that $(AB)^T = B^TA^T$
→
Find the vector equation of the plane passing through points $3\hat{\text{i}}+4\hat{\text{j}}+2\hat{\text{k}},2\hat{\text{i}}-2\hat{\text{j}}-\hat{\text{k}}$ and $7\hat{\text{i}}+6\hat{\text{k}}.$
→