Question
Evaluate the following integrals:
$\int\frac{\text{x}\sin^{-1}\text{x}^2}{\sqrt{1-\text{x}^4}}\text{ dx}$
$\int\frac{\text{x}\sin^{-1}\text{x}^2}{\sqrt{1-\text{x}^4}}\text{ dx}$
✓
Answer
Let $\text{I}=\int\frac{\text{x}\sin^{-1}\text{x}^2}{\sqrt{1-\text{x}^4}}\text{ dx}\ ....(1)$ Let $\sin^{-1}\text{x}^2=\text{t}$ then, $\text{d}\big(\sin^{-1}\text{x}^2\big)=\text{dt}$ $\Rightarrow2\text{x}\times\frac{1}{\sqrt{1-\text{x}^4}}\text{ dx}=\text{dt}$ $\Rightarrow\frac{\text{x}}{\sqrt{1-\text{x}^4}}\text{ dx}=\frac{\text{dt}}{2}$ Putting $\sin^{-1}\text{x}^2=\text{t}$ and $\frac{\text{x}}{\sqrt{1-\text{x}}^4}\text{ dx}=\frac{\text{dt}}{2}$ in equation (1),We get,
$\text{I}=\int\text{t}\frac{\text{dt}}{2}$ $=\frac{1}{2}\times\frac{\text{t}^2}{2}+\text{C}$ $=\frac{1}{4}\big(\sin^{-1}\text{x}^2\big)^2+\text{C}$ $\text{I}=\frac{1}{4}\big(\sin^{-1}\text{x}^2\big)^2+\text{C}$
$\text{I}=\int\text{t}\frac{\text{dt}}{2}$ $=\frac{1}{2}\times\frac{\text{t}^2}{2}+\text{C}$ $=\frac{1}{4}\big(\sin^{-1}\text{x}^2\big)^2+\text{C}$ $\text{I}=\frac{1}{4}\big(\sin^{-1}\text{x}^2\big)^2+\text{C}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
$\text{Evaluate:}\int\frac{\text{x + 2}}{\sqrt{\text{x}^{2}+\text{2x}+}\text{3}}\text{dx}$
→Experiments show that radium disintegrates at a rate proportional to the amount of radium present at the moment. Its half - life is 1590 years. What percentage will disappear in one year?
Let $\text{f}\text{(x)}=\frac{\log\Big(1+\frac{\text{x}}{\text{a}}\Big)-\log\Big(1-\frac{\text{x}}{\text{b}}\Big)}{\text{x}},\text{x}\neq0$ Find the value of f at x = 0. So that f becomes continuous at x = 0.
→In each of the show that the given differential equation is homogeneous and solve each of them.
$\text{y}'=\frac{\text{x}+\text{y}}{\text{x}}$
→$\text{y}'=\frac{\text{x}+\text{y}}{\text{x}}$
Find the area of the region bounded by the paraola $y^2 = 4ax$ and line $x = a.$
→Tow godowns$, A$ and $B,$ have grain storage capacity of $100$ quintals and $50$ quintals respectively. They supply to $3$ ration shops$, D, E$ and $F,$ whose requirements are $60, 50$ and $40$ quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in the following table:
How should the supplies be transported in order that the transportation cost is minimum?
→How should the supplies be transported in order that the transportation cost is minimum?
If $\text{A}=\frac{1}{9}\begin{bmatrix}-8 & 1 & 4\\4 & 4 & 7 \\ 1 & -8 & 4 \end{bmatrix},$ prove that $A^{-1} = A^3.$
→Solve the following differential equations:$\frac{\text{dy}}{\text{dx}}=\text{y}\tan2\text{x, y}(0)=2$
→$\text{Evaluate}\int\limits^{\pi}_{0} e^{2x} . \sin\big(\frac{\pi}{4} + x\big)\text{dx}$
→$\text{if y} =\text{x}^{x},\text{prove that } \frac{\text{d}^{2}\text{y}}{\text{dx}^{2}} -\frac{1}{\text{y}}\bigg(\frac{\text{dy}}{\text{dx}}\bigg)^{2} -\frac{\text{y}}{\text{x}} =0.$
→