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:

$$\int e^x(f(x) + f'(x))dx = e^x f(x) + c$$

Here $$f(x) = \sin x$$

$$\Rightarrow f'(x) = \cos x$$

$$\int e^x(\sin x + \cos x)dx = e^x \sin x + c$$

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

Formula:

$$\int e^x(f(x) + f'(x))dx = e^x f(x) + c$$

Here $$f(x) = \sec x$$

$$\Rightarrow f'(x) = \sec x \tan x$$

$$\int e^x(\sec x + \sec x \tan x)dx = e^x \sec x + c$$

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

Here $$f(x) = \log \sec x$$

$$\Rightarrow f'(x) = \tan x$$

$$\int e^x(\log \sec x + \tan x)dx = e^x \log \sec x + c$$

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

$$I = \int e^x[(1 + \tan^2 x) + \tan x] dx$$

$$= \int e^x(\sec^2 x + \tan x) dx$$

Here $$f(x) = \tan x$$

$$\Rightarrow f'(x) = \sec^2 x$$

$$\int e^x(\tan x + \sec^2 x) dx = e^x \tan x + c$$

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VSAQ-5 : Evaluate ∫e^x (1+xlogx/x)dx

Here $$f(x) = \log x$$

$$\Rightarrow f'(x) = \frac{1}{x}$$

$$\int e^x\left(1 + \frac{x \log x}{x}\right)dx$$

$$= \int e^x\left(\frac{1}{x} + \log x\right)dx = e^x \log x + c$$

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VSAQ-6 : Evaluate ∫xe^x/(x+1)²  dx\

$$I = \int e^x \frac{x e^x}{(x + 1)^2} dx = \int \left[x + 1 – \frac{1}{(x + 1)^2}\right]e^x dx$$

$$= \int e^x \left[\frac{1}{x + 1} – \frac{1}{(x + 1)^2}\right] dx$$

$$= e^x \left(\frac{1}{x + 1}\right) + c = \frac{e^x}{x + 1} + c$$

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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

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