Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluate the following integrals:$\int\frac{\text{x}^3\sin^{-1}\text{x}^2}{\sqrt{1-\text{x}^4}}\text{dx}$
✓

Answer

Let $\text{I}=\int\frac{\text{x}^3\sin^{-1}\text{x}^2}{\sqrt{1-\text{x}^4}}\text{dx}$
Let $\sin^{-1}\text{x}^2=\text{t}$
$\frac{1}{\sqrt{1-\text{x}^4}}(2\text{x})\text{dx}=\text{dt}$
$\text{I}=\int\frac{\text{x}^2\sin^{-1}\text{x}^2}{\sqrt{1-\text{x}^2}}\text{x dx}$
$=\int(\sin\text{t})\text{t}\frac{\text{dt}}{2}$
$=\frac{1}{2}\int\text{t}\sin\text{t dt}$
$=\frac{1}{2}\big[\text{t}\int\sin\text{t dt}-\int(1\int\sin\text{t dt})\text{dt}\big]$
$=\frac{1}{2}\big[\text{t}(-\cos\text{t})-\int(-\cos\text{t})\text{dt}\big]$
$=\frac{1}{2}\big[-\text{t}\cos\text{t}+\sin\text{t}\big]+\text{C}$
$\text{I}=\frac{1}{2}\Big[\text{x}^2-\sqrt{1-\text{x}^4}\sin^{-1}\text{x}^2\Big]+\text{C}$

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

Differentiate the following functions with respect to x:
$\sin^{-1}\Big\{\sqrt{\frac{1-\text{x}}{2}}\Big\},0<\text{x}<1$
→
Write the value of $\tan^{-1}\text{x}+\tan^{-1}\Big(\frac{1}{\text{x}}\Big)$ x < 0.
→
Solve the following differential equations:$\frac{\text{dy}}{\text{dx}}=\text{y}\tan2\text{x, y}(0)=2$
→
If $\text{A}=\begin{bmatrix}4 & 3 \\ 2 & 5 \end{bmatrix},$ find x and y such that $A^2 = zA + yI = 0.$ Hence, evaluate $A^{-1}.$
→
Find the area, lying above x-axis and included between the circle $x^2 + y^2 = 8x$ and the parabola $y^2 = 4x.$
→
Integrate the function $ \frac{{x + 2}}{{\sqrt {{x^2} + 2x + 3} }}$
→
Show that the four points (0, -1, -1), (4, 5, 1), (3, 9, 4) and (-4, 4, 4) are coplanar and find the equation of the common plane.
→
Evaluate the following integrals:
$\int\limits_{0}^{\pi}\frac{1}{3+2\sin\text{x}+\cos\text{x}}\text{ dx}$
→
Evaluate the following integrals:
$\int\limits^1_{-1}|2\text{x}+1|\text{dx}$
→
Solve the following differential equations:$\cos\text{y}\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}},\text{y}(0)=\frac{\pi}{2}$
→