Download App

Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
Question
Prove $\int_{0}^{1} \sin ^{-1} x d x=\frac{\pi}{2}-1$
✓

Answer

Given integral is: $\int_{0}^{1} \sin ^{-1} x d x$
To Prove: $\int_{0}^{1} \sin ^{-1} x d x=\frac{\pi}{2}-1$
Let $I=\int_{0}^{1} \sin ^{-1} x \cdot 1 d x$
because, $\int \mathrm{u} . \mathrm{v} \mathrm{dx}=\mathrm{u} \cdot \int \mathrm{v} \mathrm{dx}-\int \frac{\mathrm{d}}{\mathrm{dx}} u \cdot\left\{\int \mathrm{vdx}\right\} \mathrm{d} \mathrm{x}$
$\Rightarrow I$ = $\sin ^{-1} \mathrm{x} \cdot \int_{0}^{1} 1 \cdot \mathrm{dx}-\int_{0}^{1} \frac{\mathrm{d}}{\mathrm{dx}} \sin ^{-1} \mathrm{x} \cdot\left\{\int 1 \cdot \mathrm{dx}\right\} \cdot \mathrm{dx}$
$\Rightarrow I=\left[\sin ^{-1} \mathrm{x} \cdot \mathrm{x}\right]_{0}^{1}-\int_{0}^{1} \frac{1}{\sqrt{1-\mathrm{x}^{2}}} \cdot \mathrm{x} \mathrm{d} \mathrm{x}$
$\Rightarrow \mathrm{I}=\left[\sin ^{-1} \mathrm{x} \cdot \mathrm{x}\right]_{0}^{1}-\mathrm{I}_1$ .....(i)
First solving $I_1$:
$\Rightarrow \mathrm{I}_{1}=\int_{0}^{1} \frac{1}{\sqrt{1-\mathrm{x}^{2}}} \cdot \mathrm{x} \mathrm{d} \mathrm{x}$
Let $1 - x^2 = t \Rightarrow -2 x dx = dt$
When x = 0 then t = 1 and when x = 1 then t = 0
$\Rightarrow \mathrm{I}_{1}=\int_{1}^{0} \frac{1}{\sqrt{\mathrm{t}}} \cdot \frac{-\mathrm{dt}}{2}$
= $-\frac{1}{2}\left[\frac{t^{\frac{1}{2}}}{\frac{1}{2}}\right]_{1}^{0}$
$\Rightarrow \mathrm{I}_{1}=\sqrt{1}$
$\Rightarrow \mathrm{I}_{1}=1$
Put in equation (i)
$\Rightarrow \mathrm{I}=\left[\sin ^{-1} \mathrm{x} \cdot \mathrm{x}\right]_{0}^{1}-1$
$\Rightarrow \mathrm{I}=\sin ^{-1}(1)-0-1$
$\Rightarrow \mathrm{I}=\frac{\pi}{2}-1$
Hence Proved.

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 "1 Marks Question" 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

Area of a rectangle having vertices A, B, C and D with position vectors $ - \hat i + \frac{1}{2}\hat j + 4\hat k$, $\hat i + \frac{1}{2}\hat j + 4\hat k$, $\hat i - \frac{1}{2}\hat j + 4\hat k$ and $- \hat i - \frac{1}{2}\hat j + 4\hat k$, respectively is
→
Classify the following as scalar and vector quantities:
Distance.
→
If $f : R → R$ be defined by $f(x) = (3 - x^3)^{1/3}$​​​​​​​, then find fof(x).
→
Find the distance between the planes $2x – y + 2z = 5$ and $5x – 2.5y + 5z = 20$.
→
Evaluate the determinant $\left|\begin{array}{rrr} {2} & {-1} & {-2} \\ {0} & {2} & {-1} \\ {3} & {-5} & {0} \end{array}\right|$
→
How many arbitray constants are there in the genral solution of the differential equation of orader 3.
→
Tell that, division of a sample space is one or more.
→
Find the following integrals in Exercises: $\int(2\text{x}^2-3\sin\text{x}+5\sqrt{\text{x}})\text{ dx}$
→
$\text{If} \overrightarrow{\text{a}} = 7\hat{\text{i}} + \hat{\text{j}} - 4\hat{\text{k}}$ and $\overrightarrow{\text{b}} = 2\hat{\text{i}} - 6\hat{\text{j}} + 3\hat{\text{k}},$ then find projection of $\overrightarrow{\text{a}}\text{on} \overrightarrow{\text{b}}.$
→
If the function $\text{f(x)}=\frac{\sin10\text{x}}{\text{x}},\text{ x}\neq0$ is continuous at x = 0, find f(0).
→