Download App

Indefinite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluate the following integrals:
$\int\sqrt{1+\text{x}-2\text{x}^2}\text{dx}$
✓

Answer

Let $\text{I}=\int\sqrt{1+\text{x}-2\text{x}^2}\text{dx}$
$=\sqrt{2}\int\sqrt{\frac{1}{2}+\frac{\text{x}}{2}-\text{x}^2}\text{dx}$
$=\sqrt{2}\int\sqrt{\frac{9}{16}-\Big(\frac{1}{16}-\frac{\text{x}}{2}+\text{x}^2\Big)}\text{dx}$
$=\sqrt{2}\int\sqrt{\Big(\frac{3}{4}\Big)^2-\Big(\text{x}-\frac{1}{4}\Big)^2}\text{dx}$
$=\sqrt{2}\begin{Bmatrix}\frac{\Big(\text{x}-\frac{1}{4}\Big)}{2}\sqrt{\frac{1}{2}+\frac{\text{x}}{2}-\text{x}^2}+\frac{9}{32}\sin^{-1}\bigg(\frac{\text{x}-\frac{1}{4}}{\frac{3}{4}}\bigg)\end{Bmatrix}+\text{C}$
$\text{I}=\frac{1}{8}(4\text{x}-1)\sqrt{1+\text{x}-2\text{x}^2}+\frac{9\sqrt{2}}{32}\sin^{-1}\Big(\frac{4\text{x}-1}{3}\Big)+\text{C}$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Evaluate the following:
$\begin{bmatrix}1&-1\\0&2\\2&3\end{bmatrix}\begin{pmatrix}\begin{bmatrix}1&0&2\\2&0&1\end{bmatrix}-\begin{bmatrix}0&1&2\\1&0&2 \end{bmatrix}\end{pmatrix}$
→
Differentiate the following w.r.t. x:

$x^{e^x}+(\log x)^{\sin x}$

→
Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=\text{y}\tan\text{x}-2\sin\text{x}$
→
Evaluate the following integrals:$\int\limits^{2\pi}_0\frac{\text{e}^{\sin\text{x}}}{\text{e}^{\sin\text{x}}+\text{e}^{-\sin\text{x}}}\text{ dx}$
→
Using differentials, find the approximate values of the following:
$\sqrt{25.02}$
→
Evaluate the following:
$\begin{pmatrix}\begin{bmatrix}1&3\\-1&-4 \end{bmatrix}+\begin{bmatrix}3&-2\\-1&1 \end{bmatrix}\end{pmatrix}\begin{bmatrix}1&3&5\\2&4&6 \end{bmatrix}$
→
Prove that: $\big(\vec{\text{a}}-\vec{\text{b}}\big).\big\{\big(\vec{\text{b}}-\vec{\text{c}}\big)\big\}=0$
→
find the area of the region bounded by y = |x - 1| and y = 1.
→
Maximum Z = 3x + 4y
Subject to
$2\text{x}+2\text{y}\leq80$
$2\text{x}+4\text{y}\leq120$
→
Let R be a relation on the set A of ordered pair of integers defined by (x, y)R(u, v) if xv = yu. Show that R is an equivalence relation.
→