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Integral Calculus

Hot goIITian

 Joined: 18 Jan 2012 Post: 147
18 Feb 2012 17:26:42 IST
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Integration Challenge-1!!
Engineering Entrance , JEE Main , JEE Main & Advanced , Mathematics , Integral Calculus

Integration Challenge-1!!

I'm going to post SIX good questions on integration. The first correct solution gets a "Like" from me.

Each question will be tougher than the previous one.

Click on View post if it isn't visible.

Ok for some warm-up, comes challenge 1.

Integrate:

Blazing goIITian

Joined: 19 Jan 2008
Posts: 1142
19 Feb 2012 09:37:44 IST
2 people liked this

$\hspace{-16}\mathbf{\int_{0}^{1}e^x.\prod_{r=1}^{n}(x+r).\left(1+\sum_{r=1}^{n}\frac{1}{x+r}\right)dx}\\\\\\ \mathbf{\int_{0}^{1}e^x.\left\{\prod_{r=1}^{n}(x+r)+\prod_{r=1}^{n}(x+r).\sum_{r=1}^{n}\frac{1}{x+r}\right\}dx}\\\\\\ Now Let \mathbf{\prod_{r=1}^{n}(x+r)=f(x)}\;, Then\\\\\\ \mathbf{\prod_{r=1}^{n}(x+r).\sum_{r=1}^{n}\frac{1}{x+r}=f^{'}(x)}\\\\\\ So It Convert into \mathbf{\int_{0}^{1}e^x\left(f(x)+f^{'}(x)\right)dx=\left[e^x.f(x)\right]_{0}^{1}}\\\\\\ So \mathbf{\left[e^x.\prod_{r=1}^{n}(x+r)\right]_{0}^{1}=n!\left(en+e-1\right)}$

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