Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluate the following intergrals:
$\int\text{e}^\text{ax}\sin(\text{bx}+\text{c})\text{dx}$
✓

Answer

Let $\text{I}=\int\text{e}^\text{ax}\sin(\text{bx}+\text{c})\text{dx}$
$\Rightarrow-\text{e}^\text{ax}\frac{\cos(\text{bx}+\text{x})}{\text{b}}+\int\text{ae}^\text{ax}\frac{\cos(\text{bx}+\text{c})}{\text{b}}\text{dx}$
$=-\frac{1}{\text{b}}\text{e}^\text{ax}\cos(\text{bx}+\text{c})+\frac{\text{a}}{\text{b}}\int\text{e}^\text{ax}\cos(\text{bx}+\text{c})\text{dx}$
$=-\frac{1}{\text{b}}\text{e}^\text{ax}\cos(\text{bx}+\text{c})+\frac{\text{a}}{\text{b}}\Big[\int\text{e}^\text{ax}\frac{\sin(\text{bx}+\text{c})}{\text{b}}-\int\text{ae}^{\text{ax}}\frac{\sin(\text{bx}+\text{c})}{\text{b}}\text{dx}\Big]+\text{C}_1$
$=\frac{\text{e}^\text{ax}}{\text{b}^2}\big\{\text{a}\sin(\text{bx}+\text{c})-\text{b}\cos(\text{bx}+\text{c})\big\}\\-\frac{\text{a}^2}{\text{b}^2}\int\text{e}^\text{ax}\sin(\text{bx}+\text{c})\text{dx}+\text{C}_1$
$\Rightarrow\text{I}=\frac{\text{e}^\text{ax}}{\text{b}^2}\big\{\text{a}\sin(\text{bx}+\text{c})-\text{b}\cos(\text{bx}+\text{c})\big\}\\-\frac{\text{a}^2}{\text{b}^2}\text{I}+\text{C}_1$
$\Rightarrow\text{I}=\Big\{\frac{\text{a}^2+\text{b}^2}{\text{b}^2}\Big\}-\frac{\text{e}^\text{ax}}{\text{b}^2}\big\{\text{a}\sin(\text{bx}+\text{c})-\text{b}\cos(\text{bx}+\text{c})\big\}+\text{C}_1$
$\Rightarrow\text{I}=\frac{\text{e}^\text{ax}}{\text{a}^2+\text{b}^2}\big\{\text{a}\sin(\text{bx}+\text{c})-\text{b}\cos(\text{bx}+\text{c})\big\}+\text{C}_1$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_02\sin\text{x }\cos\text{x}\tan^{-1}(\sin\text{x})\text{dx}$
→
For what value of x the matrix A is singular?
$\text{A}=\begin{vmatrix}1+\text{x}&7\\3-\text{x}&8 \end{vmatrix}$
→
Differentiate the following functions with respect to x:
$\frac{\text{e}^\text{x}\log\text{x}}{\text{x}^2}$
→
For each of the differential equations in find a particular solution satisfying the given condition:
$(\text{x}^3+\text{x}^2+\text{x}+1) \frac{\text{dy}}{\text{dx}} = 2\text{x}^2+\text{x; y} =1 \ \text{when x = 0}$
→
Solve the following systems of linear equations by cramer's rule:
2y - 3z = 0,
x + 3y = -4,
3x + 4y = 3
→
Evaluate:$\int\limits_0^\frac{\pi}{4} \sin 2x \sin 3\text{x dx}$
→
Evaluate:
$\int\sqrt{\tan\theta}\text{ d}\theta. $
→
Prove that the function $\text{f(x)}=\begin{cases}\frac{\sin\text{x}}{\text{x}},&\text{x}<0\\\text{x}+1,&\text{x}\geq0\end{cases}$ is everywhere continuous.
→
For real numbers x and y, define xRy if $\text{x}-\text{y}+\sqrt{2}$ is an irrational number. Then the relation R is:
  1. Reflexive.
  2. Symmetric.
  3. Transitive.
  4. None of these.
→
If $\text{P}=\begin{bmatrix}\text{x}&0&0\\0&\text{y}&0\\0&0&\text{z}\end{bmatrix}$ and $\text{Q}=\begin{bmatrix}\text{a}&0&0\\0&\text{b}&0\\0&0&\text{c}\end{bmatrix},$ prove that $\text{PQ}=\begin{bmatrix}\text{xa}&0&0\\0&\text{y}\text{b}&0\\0&0&\text{zc}\end{bmatrix}=\text{QP}$
→