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Integral Calculus> Some of the toughest integrals please sol...

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1.∫(e^200x+e^202x)/(e^x+e^-x)dx.

2.∫e^x[(2-sin2x)/(1-cos2x)]dx

3.∫[x⁵]/[(x²+x+1)(x⁶+1)(x⁴-x³+x-1)]dx

4.∫[x³+2x²+x+2]/[(x²+1)^1/2]dx

5.∫dx/(5-13sinx)

6.∫dx/(a²-b²cos²x) a²>b²

DIVYA SHARMA , 12 Years ago

Grade 12th Pass

1 Answers

Jitender Singh

Last Activity: 10 Years ago

Ans:

$I = \int \frac{e^{200x} + e^{202x}}{e^{x}+e^{-x}}dx$

$I = \int e^{201x}dx = \frac{e^{201x}}{201} + constant$

$I = \int e^{x}.\frac{2-sin2x}{1-cos2x}dx$

$I = \int e^{x}.\frac{2-2sinx.cosx}{2sin^{2}x}dx$

$I = \int e^{x}.(csc^{2}x - cotx)dx$

$I = \int -e^{x}.(cotx-csc^{2}x)dx$

$I = -\int (e^{x}.cotx)'dx = -e^{x}.cotx + constant$

$I = \int \frac{x^{5}}{(x^{2}+x+1)(x^{6}+1)(x^{4}-x^{3}+x^{2}-1)}$

Simply using the partial fraction here, we have

$I = \frac{1}{12}(log(\frac{1-x^{6}}{1+x^{6}})) + constant$

$I = \frac{1}{6}tanh^{-1}(x^{6}) + constant$

$I = \int \frac{x^{3}+2x^{2}+x+1}{\sqrt{x^{2}+1}}dx$

$t = tanx$

$dt = sec^{2}x.dx$

$I = \int (sect+tan^{3}t.sect+2tan^{2}t.sect+sect.tant)dt$

$I = \frac{sec^{3}t}{3}+sect.tant + constant$

$I = \frac{1}{3}\sqrt{x^{2}+1}(x^{2}+3x+1)+constant$

$I = \int \frac{1}{5-13sinx}dx$

$t = tan(\frac{x}{2})$

$dt = \frac{1}{2}sec^{2}(\frac{x}{2})dx$

$I = 2\int \frac{1}{5t^{2}-26t+5}dt$

$I = \frac{1}{6}tanh^{-1}(\frac{1}{12}(13-5t)) + constant$

$I = \frac{1}{6}tanh^{-1}(\frac{1}{12}(13-5tan(\frac{x}{2}))) + constant$

$I = \int \frac{1}{a^{2}-b^{2}cos^{2}x}dx$

$I = \int \frac{sec^{2}x}{a^{2}sec^{2}x-b^{2}}dx$

$I = \int \frac{sec^{2}x}{a^{2}+a^{2}tan^{2}x-b^{2}}dx$

$t = tanx$

$dt = sec^{2}xdx$

$I = \int \frac{1}{a^{2}+a^{2}t^{2}-b^{2}}dt$

$I = \frac{1}{a^{2}-b^{2}}\int \frac{1}{\frac{a^{2}t^{2}}{a^{2}-b^{2}}+1}dt$

$I = \frac{tan^{-1}(\frac{at}{\sqrt{a^{2}-b^{2}}})}{a\sqrt{a^{2}-b^{2}}} + constant$

$I = \frac{tan^{-1}(\frac{atanx}{\sqrt{a^{2}-b^{2}}})}{a\sqrt{a^{2}-b^{2}}} + constant$

Thanks & Regards

Jitender Singh

IIT Delhi

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