Prove that: ^ _0xf( )dx= 2 ^ _0f( )dx - Vidyadip

Question

Prove that:
$\int\limits^\pi_0\text{xf}(\sin\text{x})\text{dx}=\frac{\pi}{2}\int\limits^\pi_0\text{f}(\sin\text{x})\text{dx}$

✓

Answer

$\int\limits^\pi_0\text{xf}(\sin\text{x})\text{dx}=\int\limits^\pi_0(\pi-\text{x})\text{f}\big[\sin(\pi-\text{x})\big]\text{dx}$ $\Bigg[\int\limits^{\text{a}}_0\text{f(x)}\text{dx}=\int\limits^{\text{a}}_0\text{f}(\text{a}-\text{x})\text{dx}\Bigg]$
$\Rightarrow\int\limits^\pi_0\text{xf}(\sin\text{x})\text{dx}=\int\limits^\pi_0(\pi-\text{x})\text{f}(\sin\text{x})\text{dx}$
$\Rightarrow\int\limits^\pi_0\text{xf}(\sin\text{x})\text{dx}=\pi\int\limits^\pi_0\text{f}(\sin\text{x})\text{dx}-\int\limits^\pi_0\text{xf}(\sin\text{x})\text{dx}$
$\Rightarrow2\int\limits^\pi_0\text{xf}(\sin\text{x})\text{dx}=\pi\int\limits^\pi_0\text{f}(\sin\text{x})\text{dx}$
$\Rightarrow\int\limits^\pi_0\text{xf}(\sin\text{x})\text{dx}=\frac{\pi}{2}\int\limits^\pi_0\text{f}(\sin\text{x})\text{dx}$

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

Show that the function f: R → {x ∈ R: –1 < x < 1} defined by $f(\text{x})=\frac{\text{x}}{1+|\text{x}|},\text{x}\in\text{R}$ is one-one and onto function.

Solve the following differential equation:
$\text{x}\frac{\text{dy}}{\text{dx}}-\text{y + x}\sin\Big(\frac{\text{y}}{\text{x}}\Big)=0$

Find the points on the curve $x^2 + y^2 = 13,$ the tangent at each one of which is parallel to the line $2x + 3y = 7.$

Evaluate: $\int\limits^{1}_{0}\cot^{-1}(1 - x + x^{2}) dx$

Prove that $x^2 – y^2 = c(x^2 + y^2)^2$ is the general solution of the differential equation $(x^3 – 3xy^2) dx = (y^3 – 3x^2y) dy,$ where $C$ is a parameter.

Find the particular solution of the following differential equation;
$\frac{\text{dx}}{\text{dy}} = 1 + x^2 + y^2 + x^2y^2, $ given that $y = 1$ when $x = 0$.

Evaluate the following integrals:
$\int\limits^{\text{b}}_{\text{a}}\frac{\text{x}^{\frac{1}{\text{n}}}}{\text{x}^\frac{1}{\text{n}}+\big(\text{a}+\text{b}-\text{x}\big)^{\frac{1}{\text{n}}}}\text{ dx},\text{ n}\in\text{N},\text{n}\leq2$

Prove that $\begin{vmatrix} \text{yz -x}^{2} & \text{zx - y}^{2} & \text{xy - z}^{2} \\ \text{zx - y}^{2} & \text{xy - z}^{2} & \text{yz - x}^{2} \\ \text{xy - z}^{2} & \text{yz - x}^{2} & \text{zx - y}^{2} \end{vmatrix}$ is divisible by (x + y + z), and hence find the quotient.

$\text{ If }\text{ x } =\cos\text{t} (3-2\cos^2\text{t)}\ \text{and}\ \text{y}=\sin\text{t}(3-2\sin\text{ t}),\text{ find the value of}\ \frac{\text{dy}}{\text{dx}} = \text{ at t}=\frac{\pi}{\text{4}}.$

Evaluate the following integrals:$\int\frac{1}{\sqrt{7-3\text{x}-2\text{x}^2}}\text{ dx}$