\(∫\frac {e^x(1+x)}{cos^2(e^x x)}\ dx \ equals\)
\(∫\frac {e^x(1+x)}{cos^2(e^x x)}\ dx \ equals\)
\(- cot (e^xx) +C\)
\(tan (xe^x) +C\)
\(tan (e^x) +C\)
\(cot (e^x) +C\)
The Correct Option is B
Solution and Explanation
\(∫\frac {e^x(1+x)}{cos^2(e^x x)}\ dx\)
Let exx = t
⇒(ex. x+ ex.1)dx = dt
ex (x+1)dx = dt
∴ \(∫\frac {e^x(1+x)}{cos^2(e^x x)}\ dx\) = \(∫\frac {dt}{cos^2 t}\)
= \(∫sec^2 t\ dt\)
= \(tan\ t+C\)
= \(tan\ (e^x. x)+C\)
= \(tan\ (xe^x)+C\)
Hence, the correct Answer is (B): \(tan\ (xe^x)+C\)
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Concepts Used:
Methods of Integration
Given below is the list of the different methods of integration that are useful in simplifying integration problems:
Integration by Parts:
If f(x) and g(x) are two functions and their product is to be integrated, then the formula to integrate f(x).g(x) using by parts method is:
∫f(x).g(x) dx = f(x) ∫g(x) dx − ∫(f′(x) [ ∫g(x) dx)]dx + C
Here f(x) is the first function and g(x) is the second function.
Method of Integration Using Partial Fractions:
The formula to integrate rational functions of the form f(x)/g(x) is:
∫[f(x)/g(x)]dx = ∫[p(x)/q(x)]dx + ∫[r(x)/s(x)]dx
where
f(x)/g(x) = p(x)/q(x) + r(x)/s(x) and
g(x) = q(x).s(x)
Integration by Substitution Method
Hence the formula for integration using the substitution method becomes:
∫g(f(x)) dx = ∫g(u)/h(u) du
Integration by Decomposition
Reverse Chain Rule
This method of integration is used when the integration is of the form ∫g'(f(x)) f'(x) dx. In this case, the integral is given by,
∫g'(f(x)) f'(x) dx = g(f(x)) + C