Mathematics

# $\int {{3^x}\cos5x\,dx = }$

$\dfrac{{{3^x}}}{{{{\left( {\log 3} \right)}^2} + 25}}\left[ {\left( {\log 3} \right).\cos5x + 5\sin5x} \right] + c$

##### SOLUTION
$I = \int {{3^x}\,\cos \,5x\,\,dx} \cdots \left( i \right)$

$I = \cfrac{{\cos 5x\,{3^x}}}{{\log 3}} + \int {\cfrac{{5\sin 5x}}{{\log 3}}dx}$

$I = \cfrac{{{3^x}\cos 5x}}{{\log 3}} + \cfrac{5}{{\log 3}}\left[ {\cfrac{{\sin 5x\,{3^{x\,}}}}{{\log 3}} - \int {\cfrac{{5\cos 5x\,{3^x}}}{{\log 3}}} \,\,dx} \right]$

$I = \cfrac{{{3^x}\cos 5x}}{{\log 3}} + \cfrac{{5\sin 5\sin x\,{3^x}}}{{{{\left( {\log 3} \right)}^2}}} - \cfrac{{25}}{{{{\left( {\log 3} \right)}^2}}}$

$I + \cfrac{{25I}}{{{{\left( {\log 3} \right)}^2}}} = \cfrac{{{3^x}\cos 5x}}{{\log 3}} + \cfrac{{5\sin 5x\,{3^x}}}{{{{\left( {log3} \right)}^2}}}$

$I = \cfrac{{25 + {{\left( {\log 3} \right)}^2}}}{{{{\left( {\log 3} \right)}^2}}}\left[ {\cfrac{{{3^x}\cos 5x}}{{\log 3}} + \cfrac{{5\sin 5x\,{3^x}}}{{{{\left( {\log 3} \right)}^2}}}} \right]$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Medium
Evaluate $\displaystyle \int \frac{x-1}{(x-3)(x-2)}dx.$
• A. $\displaystyle log(x-3)-log(x-2)$
• B. $\displaystyle 2log(x-3)+log(x-2)$
• C. $\displaystyle log(x-3)-2log(x-2)$
• D. $\displaystyle 2log(x-3)-log(x-2)$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
$\displaystyle \int_{-2}^{2}\frac{3x^{5}+4x^{3}+2x^{2}+x+20}{x^{2}+4}dx=?$
• A. $\displaystyle 3+8\pi$
• B. $\displaystyle 4+3\pi$
• C. $\displaystyle 3 +4\pi$
• D. $\displaystyle 8+3\pi$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Integrate the following w.r.t. $x$
$e^{2x} + \dfrac {1}{x^{2}}$.

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
The value of $\displaystyle \lim_{n \rightarrow \infty} \Sigma_1^n \sin\left(\dfrac{\pi}{4} +\dfrac{\pi i}{2n} \right) \dfrac{\pi}{2n}=?$
• A. $\displaystyle \int_{\tfrac{\pi}{2}}^{\tfrac{\pi}{4}} \sin x dx$
• B. $\displaystyle \int_{\tfrac{\pi}{2}}^{\tfrac{3\pi}{4}} \sin x dx$
• C. $\displaystyle \int_\pi^{3\pi} \sin x dx$
• D. $\displaystyle \int_{\tfrac{\pi}{4}}^{\tfrac{3\pi}{4}} \sin x dx$

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard

In calculating a number of integrals we had to use the method of integration by parts several times in succession. The result could be obtained more rapidly and in a more concise form by using the so-called generalized formula for integration by parts.

$\int u(x)\, v(x)dx\, =\, u(x)\, v_{1}(x)\, -\, u^{}(x)v_{2}(x)\, +\, u^{}(x)\, v_{3}(x)\, -\, .\, +\, (-1)^{n\, -\, 1}u^{n\, -\, 1}(x)v_{n}(x)\, -\, (-1)^{n\, -\, 1}$ $\int\, u^{n}(x)v_{n}(x)\, dx$ where $v_{1}(x)\, =\, \int v(x)dx,\, v_{2}(x)\, =\, \int v_{1}(x)\, dx\, ..\, v_{n}(x)\, =\, \int v_{n\, -\, 1}(x) dx$

Of course, we assume that all derivatives and integrals appearing in this formula exist. The use of the generalized formula for integration  by parts is especially useful when calculating $\int P_{n}(x)\, Q(x)\, dx$, where $P_{n}(x)$, is polynomial of degree n and the factor Q(x) is such that it can be integrated successively n + 1 times.