Evaluate the following integrals: 4x^3 5-x^2 dx - Vidyadip

Question

Evaluate the following integrals:
$\int4\text{x}^3\sqrt{5-\text{x}^2}\text{ dx}$

✓

Answer

$\int4\text{x}^3\sqrt{5-\text{x}^2}\text{ dx}$
$=4\int\text{x}^2\times\text{x}\sqrt{5-\text{x}^2}\text{ dx}$
Let $5-\text{x}^2=\text{t}$
$\Rightarrow\text{x}^2=5-\text{t}$
$\Rightarrow2\text{x}=-\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\text{x dx}=-\frac{\text{dt}}{2}$
Now, $4\int\text{x}^2\times\text{x}\sqrt{5-\text{x}^2}\text{ dx}$
$=\frac{4}{-2}\int(5-\text{x})\sqrt{\text{t}}\text{ dt}$
$=-2\int5\text{t}^{\frac{1}{2}}+2\int\text{t}^{\frac{3}{2}}\text{ dt}$
$=-10\Bigg[\frac{\text{t}^{\frac{1}{2}+1}}{\frac{1}{2}+1}\Bigg]+2\Bigg[\frac{\text{t}^{\frac{3}{2}+1}}{\frac{3}{2}+1}\Bigg]+\text{C}$
$=-\frac{20}{3}\text{t}^{\frac{3}{2}}+\frac{4}{5}\text{t}^{\frac{5}{2}}+\text{C}$
$=-\frac{20}{3}\big(5-\text{x}^2\big)^{\frac{3}{2}}+\frac{4}{5}\big(5-\text{x}^2\big)^{\frac{5}{2}}+\text{C}$

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 "5 Marks Questions" from INTEGRALS→INTEGRALS — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

If R is a relation on the set A = {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3)}, then R is:

Reflexive.
Symmetric.
Transitive.
All the three options.

Two cards are drawn simultaneosly from a well shuffled deck of 52 cards. Find the probability distribution of the number of the successes, when getting a spade is considered a success.

Solve the following initial value problems:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{y}(\text{x}+2\text{y})}{\text{x}(2\text{x}+\text{y})},\text{y}(1)=2$

If a young man drives his vehicle at 25 km/hr, he has to spend Rs. 2 per km on petrol. If he drives it at a faster speed of 40 km/hr, the petrol cost increases to Rs. 5 per km. He has Rs. 100 to spend on petrol and travel within one hour. Express this as an LPP and solve the same.

Find the second order derivatives of the following functions:
$\text{y}=\tan^{-1}\text{x}$

Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}-\text{x}\text{e}^\text{x}-\frac{5}{2}+\cos^2\text{x}$

Solve the following systems of linear equations by cramer's rule:

2x - 3z + w = 1,

x - y + 2w = 1,

-3y + z + w = 1,

Prove that:
$\begin{vmatrix}\text{z}&\text{x}&\text{y}\\\text{z}^2&\text{x}^2&\text{y}^2\\\text{z}^4&\text{x}^4&\text{y}^4 \end{vmatrix}=\begin{vmatrix}\text{x}&\text{y}&\text{z}\\\text{x}^2&\text{y}^2&\text{z}^2\\\text{x}^4&\text{y}^4&\text{z}^4 \end{vmatrix}=\begin{vmatrix}\text{x}^2&\text{y}^2&\text{z}^2\\\text{x}^4&\text{y}^4&\text{z}^2\\\text{x}&\text{y}&\text{z}\end{vmatrix}$
$=\text{xyz}(\text{x}-\text{y})(\text{y}-\text{z})(\text{z}-\text{x})(\text{x}+\text{y}+\text{z}).$

Differentiate the following functions with respect to x:
$(\sin^{-1}\text{x})^\text{x}$

Calculate the area of the ragion bounded by $y^2= x$ and $x^2= y.$