29 Most VSAQ’s of Integration Chapter in Inter 2nd Year Maths-2B (TS/AP)

2 Marks

VSAQ-1 : Evaluate ∫e^x(sinx+cosx)dx

Formula:

∫ex(f(x)+f′(x))dx=exf(x)+c

Here

f(x)=sinx

⇒f′(x)=cosx

∫ex(sinx+cosx)dx=exsinx+c

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VSAQ-2 : Evaluate ∫e^x(secx+secxtanx)dx

Formula:

∫ex(f(x)+f′(x))dx=exf(x)+c

Here

f(x)=secx

⇒f′(x)=secxtanx

∫ex(secx+secxtanx)dx=exsecx+c

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VSAQ-3 : Evaluate ∫e^x(tanx+log⁡ secx)dx

Here

f(x)=logsecx

⇒f′(x)=tanx

∫ex(logsecx+tanx)dx=exlogsecx+c

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VSAQ-4 : Evaluate ∫e^x (1 + tan² x + tanx)dx

I=∫ex[(1+tan2x)+tanx]dx

=∫ex(sec2x+tanx)dx

Here

f(x)=tanx

⇒f′(x)=sec2x

∫ex(tanx+sec2x)dx=extanx+c

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VSAQ-5 : Evaluate ∫e^x (1+xlogx/x)dx

Here

f(x)=logx

⇒f′(x)=1x

∫ex(1+xlogxx)dx

=∫ex(1x+logx)dx=exlogx+c

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VSAQ-6 : Evaluate ∫xe^x/(x+1)²  dx\

I=∫exxex(x+1)2dx=∫[x+1–1(x+1)2]exdx

=∫ex[1x+1–1(x+1)2]dx

=ex(1x+1)+c=exx+1+c

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VSAQ-7 : Evaluate ∫(1/1-x²+1/1+x²)dx

∫(1/1 – x² + 1/1 + x²)ndx = ∫dx/1 – x² + ∫dx/1 + x²

= Tanh^-1x + Tan^-1x + x

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VSAQ-8 : Evaluate ∫dx/(x+1)(x+2)

∫dx/(x + 1)(x + 2) = ∫(x + 2) – (x + 1)/(x + 1)(x + 2)

= ∫(1/x + 1 – 1/x + 2)dx = log|x + 1| – log|x + 2| + c

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VSAQ-9 : Evaluate ∫sec² x.csc² xdx

∫sec² x.csc² xdx = ∫1/cos² x sin² x dx

= ∫sin²x + cos²x/cos²x sin²x dx

= ∫(1/cos²x + 1/sin²x)dx

= ∫(sec²xdx + ∫csc² xdx

= ∫sec² xdx + ∫csc² xdx

= tan x – cot x + c

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VSAQ-10 : Evaluate ∫1/(coshx+sinhx) dx

∫1/coshx + sinhx dx

= ∫cosh²x – sinh²x/coshx + sinhx dx

= ∫(cos hx – sin hx)(cos hx + sin hx)/coshx + sinhx

= ∫(cos hx – sin hx)dx = sin hx – cos hx + c