Evaluate the following integrals: _ 0 ^ 2 1+ ^4x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits_{0}^{\frac{\pi}{2}}\frac{\sin\text{x}\cos\text{x}}{1+\sin^4\text{x}}\text{ dx}$

✓

Answer

Let $\sin^2\text{x}=\text{t}$
Differentiating w.r.t. x, we get
$2\sin\text{x}\cos\text{x dx}=\text{dt}$
Now, $\text{x}=0\Rightarrow\text{t}=0$
$\text{x}=\frac{\pi}{2}\Rightarrow\text{t}=1$
$\int_{0}^\limits{\frac{\pi}{2}}\frac{\sin\text{x}\cos\text{x}}{1+\sin^4\text{x}}\text{ dx}$
$=\frac{1}{2}\int\limits^1_0\frac{\text{dt}}{1+\text{t}^2}$
$=\frac{1}{2}\big[\tan^{-1}\text{t}\big]^1_0$
$=\frac{1}{2}\Big[\tan^{-1}(1)-\tan^{-1}(0)\Big]$
$=\frac{1}{2}\Big[\tan^{-1}\Big(\tan\frac{\pi}{4}\Big)-\tan^{-1}(\tan0)\Big]$
$=\frac{1}{2}\times\frac{\pi}{4}$
$=\frac{\pi}{8}$

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 Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find a particular solution of the differential equation $\frac{ dy }{ dx }+2 y \cot x=4 x \operatorname{cosec} x$ $(x \neq 0)$, given that $y=0$, when $x=\frac{\pi}{2}$.

Find the direction cosines of the sides of the triangle whose vertices are $(3, 5, -4), (-1, 1, 2)$ and $(-5, -5, -2).$

Using vector method, prove that the point is collinear:
A(6, -7, -1), B(2, -3, 1) and C(4, -5, 0)

Solve the Linear Programming Problem graphically:
Minimize Z = 3x + 5y such that $x + 3y \geq 3,x + y \geq 2,x,y \geq 0$.

Using vectors, find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).

Solve the following differential equation
$\sqrt{\text{a}+\text{x}}\text{dy}+\text{x dx}=0$

Prove that $\Big(\frac{\text{x}}{\text{a}}\Big)^\text{n}+\Big(\frac{\text{y}}{\text{b}}\Big)^\text{n}=2$ touches the straight line $\frac{\text{x}}{\text{a}}+\frac{\text{y}}{\text{b}}=2$ for all n $\in$ N, at all the point (a, b).

Find the reflection of the point (1, 2, -1) in the plane 3x - 5y + 4z = 5.

Show that the points whose position vectors are as given below are collinear:
$2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}},\ 3\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}$ and $\hat{\text{i}}+4\hat{\text{j}}-3\hat{\text{k}}$

Two numbers are selected at random from integers 1 through 9. If the sum is even, find the probability that both the numbers are odd.