Question
Evaluate the following integrals:
$\int_{0}^\limits{1}\tan^{-1}\text{x dx}$
$\int_{0}^\limits{1}\tan^{-1}\text{x dx}$
✓
Answer
Let $\text{I}=\int_{0}^\limits{1}\tan^{-1}\text{x dx}$ Then,
$\text{I}=\int_{0}^\limits{1}\tan^{-1}\text{x dx}$
Integrating by parts,
$\text{I}=\big[\text{x}\tan^{-1}\text{x}\big]^1_0-\int_{0}^\limits{1}\frac{\text{x}}{1+\text{x}^2}\text{ dx}$
$\Rightarrow\text{I}=\big[\text{x}\tan^{-1}\text{x}\big]^1_0-\frac{1}{2}\big[\log\big(\text{x}^2+1\big)\big]^1_0$
$\Rightarrow\text{I}=\frac{\pi}{4}-0-\frac{1}{2}\log2+0$
$\Rightarrow\text{I}=\frac{\pi}{4}-\frac{1}{2}\log2$
$\text{I}=\int_{0}^\limits{1}\tan^{-1}\text{x dx}$
Integrating by parts,
$\text{I}=\big[\text{x}\tan^{-1}\text{x}\big]^1_0-\int_{0}^\limits{1}\frac{\text{x}}{1+\text{x}^2}\text{ dx}$
$\Rightarrow\text{I}=\big[\text{x}\tan^{-1}\text{x}\big]^1_0-\frac{1}{2}\big[\log\big(\text{x}^2+1\big)\big]^1_0$
$\Rightarrow\text{I}=\frac{\pi}{4}-0-\frac{1}{2}\log2+0$
$\Rightarrow\text{I}=\frac{\pi}{4}-\frac{1}{2}\log2$
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
In Figure ABCD is a regular hexagon, which vectors are:
- Collinear.
- Equal.
- Co-initial.
- Collinear but not equal.
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
$\text{f}(\text{x})=\sqrt{25-\text{x}^2}\text{ on }[-3,4]$
→$\text{f}(\text{x})=\sqrt{25-\text{x}^2}\text{ on }[-3,4]$
Prove that the diagonals of a rectangle are perpendicular if and only if the rectangle is a square.
Show that $\text{f}(\text{x})=\sin\text{x}-\cos\text{x}$ is an increasing function on $\Big(-\frac{\pi}{4},\frac{\pi}{4}\Big).$
→If R and S are relations on a set A, then prove that:
→- R and S are symmetric $\Rightarrow\ \text{R}\cap\text{S}$ and $\text{R}\cup\text{S}$ are symmetric
- R is reflexive and S is any relation $\Rightarrow\ \text{R}\cup\text{S}$ is reflexive.
If $\text{A}=\begin{bmatrix}\text{ab}&\text{b}^2\\-\text{a}^2&-\text{ab}\end{bmatrix},$ show that $A^2 = 0$
→Evaluate the following integrals:$\int\text{e}^{\text{x}}\frac{(1-\text{x})^2}{(1+\text{x}^2)^2}\text{dx}$
→Solve the following differential equation
$\text{xy}(\text{y}+1)\text{dy}=(\text{x}^2+1)\text{dx}$
→$\text{xy}(\text{y}+1)\text{dy}=(\text{x}^2+1)\text{dx}$
A metal box with a square base and vertical sides is to contain $1024\ cm^3.$ The material for the top and bottom costs $₹\ 5\ per\ cm^2 $ and the material for the sides costs $₹\ 2.50\ per\ cm^2.$ Find the least cost of the box.
→Find the area of the ragion bounded by $x^2 = 16y, y = 1, y = 4$ and the parabola y-axis and the first quadrant.
→