Download App

Indefinite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
$\int\frac{\text{x}^2}{\sqrt{3\text{x}-4}}\text{dx}$
✓

Answer

$\int\frac{\text{x}^2}{\sqrt{3\text{x}-4}}\text{dx}$
Let $3\text{x}+4=\text{t}$
$\Rightarrow\text{x}=\frac{\text{t}^{-4}}{3}$
$\Rightarrow1=\frac{1}{3}\cdot\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\text{dx}=\frac{\text{dt}}{3}$
Now, $\int\frac{\text{x}^2}{\sqrt{3\text{x}-4}}$
$=\frac{1}{3}\int\frac{\Big(\frac{\text{t}^{-4}}{3}\Big)^2}{\sqrt{\text{t}}}\text{dt}$
$=\frac{1}{27}\int\Big(\frac{\text{t}^2}{\sqrt{\text{t}}}-\frac{8\text{t}}{\sqrt{\text{t}}}+\frac{16}{\sqrt{t}}\Big)\text{dt}$
$=\frac{1}{27}\int\Big(\text{t}^\frac{3}{2}-8\text{t}^\frac{1}{2}+16\text{t}^{-\frac{1}{2}}\Big)\text{dt}$
$=\frac{1}{27}\Bigg[\frac{\text{t}^{\frac{3}{2}+1}}{\frac{3}{2}+1}+\frac{8\text{t}^{\frac{1}{2}+1}}{\frac{1}{2}+1}+\frac{16\text{t}^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}\Bigg]+\text{C}$
$=\frac{1}{27}\Big[\frac{2}{5}\text{t}^{\frac{5}{2}}-\frac{8\times2}{3}\text{t}^{\frac{3}{2}}+32\text{t}^{\frac{1}{2}}\Big]+\text{C}$
$=\frac{2}{135}(\text{t})^{\frac{5}{2}}-\frac{16}{81}\text{t}^{\frac{3}{2}}+\frac{32}{27}\text{t}^\frac{1}{2}+\text{C}$
$=\frac{2}{135}(3\text{x}+4)^{\frac{5}{2}}-\frac{16}{81}(3\text{x}+4)^{\frac{3}{2}}+\frac{32}{27}(3\text{x}+4)^{\frac{1}{2}}+\text{C}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

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

Find the position vector of point R which divides the line joining the points P and Q whose

position vectors are $2 \hat{i}-\hat{j}+3 \hat{k}$ and $-5 \hat{i}+2 \hat{j}-5 \hat{k}$ in the ratio $3: 2$

(1)internally

(2)externally

→
Evaluate the following integrals as limit of sum:$\int\limits^2_{0}\text{e}^{\text{x}}\text{ dx}$
→
Sketch the region ${(x, y): 9x^2+ 4y^2 = 36}$ and find the area of the enclosed by it, using integration.
→
Show that $\text{y}=\frac{\text{c}-\text{x}}{1+\text{cx}}$ is a solution of the differential equation $(1+\text{x}^2)\frac{\text{dy}}{\text{dx}}+(1+\text{y}^2)=0.$
→
Solve the following system of equations by matrix method:
$5x +3y + z = 16$
$2x + y +3z = 19$
$x + 2y + 4z = 25$
→
Solve the following differential equations:$\frac{\text{dy}}{\text{dx}}=2\text{e}^{2\text{x}}\text{y}^2,\text{y}(0)=-1$
→
Prove that a conical tent of given capacity will require the least amount of canvas when the height is √2 times the radius of the base.
→
Evaluate the following definite integrals:$\int_{0}^\limits{1}\Big(\text{xe}^{\text{x}}+\cos\frac{\pi\text{x}}{4}\Big)\text{dx}$
→
For any a, b, x, y > 0, prove that:
$\frac{2}{3}\tan^{-1}\Big(\frac{3\text{a}\text{b}^2-\text{a}^3}{\text{b}^3-3\text{a}^2\text{b}}\Big)+\frac{2}{3}\tan^{-1}\Big(\frac{3\text{x}\text{y}^2-\text{x}^3}{\text{y}^3-3\text{x}^2\text{y}}\Big)=\tan^{-1}\frac{2\alpha\beta}{\alpha^2-\beta^2}$
where $\alpha=-\text{ax}+\text{by},\beta=\text{bx}+\text{ay}$
→
Evaluate the following integrals:$\int\limits^{\infty}_0\frac{\text{x}}{(1+\text{x})(1+\text{x}^2)}\text{ dx}$
→