Evaluate: e^x(1+x) ^2 (x e^x ) d x - Vidyadip

Question

Evaluate: $\int \frac{e^x(1+x)}{\cos ^2\left(x e^x\right)} d x$

✓

Answer

Let $I =\int \frac{e^x(1+x)}{\cos ^2\left(x e^x\right)} d x$
Put $x.e^x = t$
Diff.$w.r.t..x$
$ \therefore x \cdot e^x+e^x \cdot 1=\frac{d t}{d x}$
$ \therefore e^x(1+x) d x=d t$
$ \therefore I=\int \frac{d t}{\cos ^2 t}=\int \sec ^2 t d t$
$ \therefore \tan t+c$
$ =\tan \left( x \cdot e ^{ x }\right)+ e $

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.(2 Marks)" from Indefinite Integration→Indefinite Integration — 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 integrals:$\int\frac{\cos\text{x}}{\sqrt{4-\sin^2\text{x}}}\text{ dx}$

Write the domain of the real function $\text{f(x)}=\sqrt{[\text{x}]-\text{x}}.$

In $\triangle ABC$, prove that $a (\cos C -\cos B )=2( b - c ) \cos ^2\left(\frac{A}{2}\right)$

Draw the graph of inequalities $x ≤ 6, y −2 ≤ 0, x ≥ 0, y ≥ 0$ and indicate the feasible region

Obtain the probability distribution of the number of sixes in two tosses of a fair die.

Define a commutative binary operation on a set.

If $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are position vectors of the points A, B and C respectively, write the value of$\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}+\overrightarrow{\text{AC}}$.

$\bar{u} \cdot \bar{v}$ if $|\bar{u}|=2,|\vec{v}|=5,|\bar{u} \times \bar{v}|=8$

Evaluate the following integrals:
$\int\frac{\sin\sqrt{\text{x}}}{\sqrt{\text{x}}}\text{ dx}$

Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}+2\text{x}=\text{e}^{3\text{x}}$