If u and v are two functions of x- then prove that- - u v dx - u- v dx - - -dudx- v dx - dx

It is given that u and v are two functions of x .Assume, ∫ v dx = V And, dudx = u' We know that (uV) #8217; = uv + u #8217;V I ...

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Standard XII

Mathematics

Integration by Parts

If u and ...

Question

If $u$ and $v$ are two functions of $x$ , then prove that:
$\int u v d x = u \int v d x - \int [\frac{d u}{d x} \int v d x] d x$

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Solution

It is given that $u$ and $v$ are two functions of $x$ .
Assume,
$\int v d x = V$
And, $\frac{d u}{d x} = u^{'}$

We know that
$(u V)^{'} = u v + u^{'} V$
If we integrate both sides, we get
$\int (u V)^{'} d x = \int u v d x + \int u^{'} V d x$
$\Rightarrow u V = \int u v d x + \int u^{'} V d x$
Or,
$\int u v d x = u V - \int u^{'} V d x$
Therefore,
$\int u v d x = u \int v d x - \int [\frac{d u}{d x} \int v d x] d x$

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If u and v are two functions of x, then prove that:∫u v dx=u∫v dx−∫[dudx∫v dx]dx

It is given that u and v are two functions of x.Assume,∫v dx=VAnd, dudx=u′We know that(uV)′=uv+u′VIf we integrate both sides, we get∫(uV)′ dx=∫uv dx+∫u′V dx⇒uV=∫uv dx+∫u′V dxOr,∫uv dx=uV−∫u′V dxTherefore,∫u v dx=u∫v dx−∫[dudx∫v dx]dx

If $u$ and $v$ are two functions of $x$ , then prove that:
$\int u v d x = u \int v d x - \int [\frac{d u}{d x} \int v d x] d x$