Evaluate x+ 6x 3x^2+ 6x dx - Vidyadip

Question

Evaluate $\int\frac{\text{x}+\cos6\text{x}}{3\text{x}^2+\sin6\text{x}}\text{ dx}$

✓

Answer

Let $3\text{x}^2+\sin6\text{x}=\text{t}$
$6\text{x}+6\cos6\text{x dx}=\text{dt}$
$(\text{x}+\cos6\text{x})\text{dx}=\frac{\text{dt}}{6}$
Thus, $\text{I}=\int\frac{\text{x}+\cos6\text{x}}{3\text{x}^2+\sin6\text{x}}\text{ dx}$
$=\frac{1}{6}\int\frac{\text{dt}}{\text{t}}$
$=\frac{1}{6}\log|\text{t}|+\text{C}$
$=\frac{\log\big|3\text{x}^2+\sin6\text{x}\big|}{6}+\text{C}$

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 "2 Marks" from INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Let '*' be a binary operation on N defined by a * b = 1.c.m. (a, b) for all $\text{a, b}\in\text{N}$
Find 2 * 4, 3 * 5, 1 * 6.

Write the value of k for which the line $\frac{\text{x}-1}{2}=\frac{\text{y}-1}{3}=\frac{\text{z}-1}{\text{k}}$ is perpendicular to the normal to the plane $\vec{\text{r}}.(2\hat{\text{i}}+3\hat{\text{j}}+4\hat{\text{k}})=4.$

For what value of x, is the matrix $\text{A}=\begin{bmatrix}0&1&-2\\-1&0&3\\\text{x}&-3&0 \end{bmatrix}$ a skew-symmetric matrix?

Let $\text{f}:\Big(-\frac{\pi}{2},\frac{\pi}{2}\Big)\rightarrow\ \text{R}$ be a function defined by f(x) = cos[x]. write range (f).

Find the general solution of the differential equation $\frac{d y}{d x}=\sin ^{-1} x$

A company produces two types of goods A and B, that require gold and silver. Each unit of type A requires 3 g of silver and 1 g of gold while that of type B requires 1 g of silver and 2 g of gold. The company can procure a maximum of 9 g of silver and 8 g of gold. If each unit of type A brings a profit of ₹ 40 and that of type B ₹ 50, formulate LPP to maximize profit.

If $A=\left[\begin{array}{c}-2 \\ 4 \\ 5\end{array}\right], B=\left[\begin{array}{lll}1 & 3 & 6\end{array}\right]$, then show that $(A B)^T=B^T \cdot A^T$

If x and y are connected parametrically by the equation x = a ($\theta$ – sin $\theta$), y = a (1 + cos $\theta$) without eliminating the parameter. Find $\frac{d y}{d x}$.

Evaluate: $\begin{vmatrix}\cos15^\circ&\sin15^\circ\\\sin75^\circ&\cos75^\circ\end{vmatrix}$

What is the maximum value of $a \sin x+b \cos x$ ?