Question
Find: $\int \frac{d x}{\sqrt{x}+x}$
✓
Answer
$\int \frac{d x}{\sqrt{x}+x} d x=\int \frac{1}{\sqrt{x}(1+\sqrt{x})} d x$
Let, $1+\sqrt{x}=t$
$\begin{aligned} 0+\frac{1}{2 \sqrt{x}} d x & =d t \\ \frac{1}{\sqrt{x}} d x & =2 d t \\ & =2 \int \frac{1}{t} d t \\ & =2 \log t+C \\ & =2 \log (1+\sqrt{x})+C\end{aligned}$
Let, $1+\sqrt{x}=t$
$\begin{aligned} 0+\frac{1}{2 \sqrt{x}} d x & =d t \\ \frac{1}{\sqrt{x}} d x & =2 d t \\ & =2 \int \frac{1}{t} d t \\ & =2 \log t+C \\ & =2 \log (1+\sqrt{x})+C\end{aligned}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Form the differential equation representing the family of curves $y=e^{2 x}(a+b x)$where 'a' and 'b' are arbitrary constants.
→A container contains $70 /$ of orange squash. The squash being too concentrated $7 /$ of squash was taken out from this container and replaced by water. This process was repeated thrice to reduce the concentration of squash. How much quantity of orange squash is left in the container?
→What time will it be after 200 hours, if the present time is 5:00 am?
If x (mod 9)=2, find all the possible values of x; where 0 < x < 47.
What is the relationship between arrival time, burst time, completion time and waiting time?
Mr. Roy purchased a home for ₹ 200,000 in 2012. In the next few years, homes in neighbourhood have been selling well due to the new shopping plaza a couple of miles away, which increased the market value of his home. So in 2017, he decided to downsize and sell his home. Based on the current market value during this time, he was able to sell his home for ₹ 250,000. Find his rate of return.
How many kg of sugar costing ₹ 45 per kg must be mixed with 30 kg sugar costing ₹ 35 per kg so that there may be a gain of 12% by selling the mixture at ₹ 47.04 per kg?
For what value of $x$, the given matrix $A=\left|\begin{array}{cc}3-2 x & x+1 \\ 2 & 4\end{array}\right|$ is a singular matrix?
→What is the area bounded by the curve y = log x, X-axis and the ordinates x = 1, x = 2 ?