Download App

Indefinite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluate the following integrals:
$\int\text{x}^2\text{e}^{\text{x}^3}\cos\text{x}^3\text{dx}$
✓

Answer

Given integral is,
$\text{I}=\int\text{x}^2\text{e}^{\text{x}^3}\cos(\text{x}^3)\text{dx}$
Let $x^3 = t$
$\Rightarrow3\text{x}^2\text{dx}=\text{dt}$
$\Rightarrow\text{x}^2\text{dx}=\frac{\text{dt}}{3}$
Integral becomes,
$\frac{1}{3}\int\text{e}^\text{t}\cos\text{t dt}$
$=\frac{1}{3}\text{I}\ \dots(1)$
Where, $\text{I}=\int\text{e}^\text{t}\cos\text{t dt}$
$\text{I}=\int\text{e}^\text{t}\cos\text{t dt}$
Considering cos t as first and $e^t$ as second function
$\text{I}=\cos\text{t e}^\text{t}-\int-\sin\text{t e}^\text{t}\text{dt}$
$\Rightarrow\text{I}=\text{e}^\text{t}\cos\text{t}+\int\sin\text{t }\text{e}^\text{t}\text{dt}$
Again considering sin t as first and $e^t$ as second function
$\text{I}=\text{e}^\text{t}\cos\text{t}+\sin\text{t }\text{e}^\text{t}-\int\cos\text{t e}^\text{t}\text{dt}$
$\Rightarrow\text{I}=\text{e}^\text{t}\cos\text{t}+\sin\text{t e}^\text{t}-1$
$\Rightarrow2\text{I}=\text{e}^\text{t}(\sin\text{t}+\cos\text{t})$
$\Rightarrow\text{I}=\frac{\text{e}^\text{t}}{2}(\sin\text{t}+\cos\text{t})$
$\therefore\ \int\text{x}^2\text{e}^{\text{x}^3}\cos(\text{x}^3)\text{dx}=\frac{1}{3}\Big[\frac{\text{e}^\text{t}}{2}(\sin\text{t}+\cos\text{t})\Big]+\text{C}$ [From (1)]
$=\frac{\text{e}^{\text{x}^3}}{6}(\sin\text{x}^3+\cos\text{x}^3)+\text{C}$

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

Differentiate the following w.r.t. x:

$\left[(\tan x)^{\tan x}\right]^{\tan x}$ at $x=\frac{\pi}{4}$

→
A firm has to transport at least 1200 packages daily using large vans which carry 200 packages each and small vans which can take 80 packages each. The cost of engaging each large van is Rs 400 and each small van is Rs 200. Not more than Rs 3000 is to be spent daily on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as a LPP given that the objective is to minimize cost.
→
Evaluate the following integrals:$\int\frac{2\text{x}+5}{\text{x}^2-\text{x}-2}\text{ dx}$
→
Solve the following determinant equations:
$\begin{vmatrix}3\text{x}-8&3&3\\3&3\text{x}-8&8\\3&3&3\text{x}-8\end{vmatrix}=0$
→
Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{y}^2=4\text{x}\text{ at }(1,2)$
→
Find the equations of tangents and normals to the curve at the point on it.

$x \sin 2 y=y \cos 2 x$ at $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$

→
Differentiate $\tan ^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ w.r.t. $\cos ^{-1}\left(\sqrt{\frac{1+\sqrt{1+x^2}}{2 \sqrt{1+x^2}}}\right)$
→
Show that the following system of linear equation is inconsistent:
x + y − 2z = 5
x − 2y + z = −2
−2x + y + z = 4
→
Evaluate the following integrals:
$\int4\text{x}^3\sqrt{5-\text{x}^2}\text{ dx}$
→
Evaluate the following determinant:
$\begin{vmatrix}1&3&9&27\\3&9&27&1\\9&27&1&3\\27&1&3&9 \end{vmatrix}$
→